5th international conference on computational harmonic ...for the algebraic problem, we de ne a...

5th International Conference on Computational

Harmonic Analysis

Vanderbilt University, Department of MathematicsNashville, TN, USA

May 19–23, 2014

Variable bandwidth in directional time-frequency analysisRoza AceskaVanderbilt [email protected]

We introduce the notion of variable bandwidth in time-frequency analysiswith the additional element of direction. The directional short-time Fouriertransform provides for a directionally sensitive time-frequency analysis. In ad-dition, we define a directionally sensitive variable bandwidth weight on thetime-frequency plane, which controls the frequency decay of a function (e.g.image) in a certain time interval and in a chosen direction.

Getting even more from less: Structure-exploiting sampling strategiesfor compressive imagingBen AdcockPurdue [email protected]

Coauthors: Anders C. Hansen, Clarice Poon and Bogdan Roman

It is well-known that natural images are not just sparse in wavelet basesand their various generalizations, but that there is a distinct structure to thissparsity. In this talk we present a method and a full theory for leveragingsuch structure within the context compressed sensing. This approach is basedon recent theoretical developments which show that compressed sensing is pos-sible under the substantially relaxed condition of asymptotic incoherence, asopposed to uniform incoherence. Structured sparsity can be exploited by sam-pling with asymptotically incoherent transforms according to a so-called mul-tilevel random subsampling strategy. This approach improves on conventionalcompressed sensing strategies - typically based uniformly-incoherent randomGaussian and Bernoulli sensing matrices - in two important ways. First, oneobtains a higher reconstruction quality, since uniformly incoherent matrices can-not exploit structured sparsity. Second, since it is a much weaker condition thanuniform incoherence, one can easily find computationally-efficient sensing ma-trices that are asymptotically incoherent with wavelet bases. In particular, theFourier and Hadamard transforms have this property, and unlike random ma-trices, both are highly efficient in terms of storage and computational cost. Inthe final part of the talk, we also compare this approach with other structure-exploiting algorithms for compressed sensing (typically based on wavelet tree orGaussian mixture models) which seek to exploit structure by suitable modifica-tion of the reconstruction algorithm. We show that our approach of ‘structuredsampling’, which uses the standard reconstruction algorithm of l1 minimization,also outperforms these ‘structured recovery’ algorithms both in terms of speedand accuracy.

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Interval Linear Optimization with Fuzzy Inequality ConstraintsIbraheem AlolyanKing Saud [email protected]

In many real-life situations, we come across problems with imprecise inputvalues. Imprecisions are dealt with by various ways. One of them is intervalbased approach in which we model imprecise quantities by intervals, and supposethat the quantities may vary independently and simultaneously within theirintervals. In most optimization problems, they are formulated using impreciseparameters. Such parameters can be considered as fuzzy intervals, and theoptimization tasks with interval cost function are obtained. When realisticproblems are formulated, a set of intervals may appear as coefficients in theobjective function or the constraints of a linear programming problem. In thispaper, we introduce a new method for solving linear optimization problems withinterval parameters in the objective function and the inequality constraints,and we show the efficiency of the proposed method by presenting a numericalexample.

Sampling and interpolation sets near the critical density in LCAgroupsJorge Abel [email protected]

Coauthors: E. Agora, C. Cabrelli

In [3] Landau found necessary conditions satisfied by sampling and inter-polation sets for Paley Wiener spaces PWK , associated to a bounded set K ofRd. These conditions, given in terms of the so called lower and upper Beurlingdensities, D− and D+ respectively, are the following:

i. A sampling set L for PWK satisfies D−(L) ≥ |K|;ii. An interpolation set L for PWK satisfies D+(L) ≤ |K|.A natural question is whether or not there exist sampling and interpolation

sets for PWK with densities arbitrarily close to the critical density |K|. This istrue, and it was proved by Marzo in [6] in Rd, adapting ideas of Lyubarskii andSeip [4] and Kohlenberg [2]. It also follows from a more recent work by Mateiand Meyer [5].

Later on, in [1], Grochenig, Kutyniok, and Seip suitably extended the con-cept of Beurling densities to the setting of locally compact abelian groups (LCAgroups), and they proved a generalization of aforementioned Landau’s theorem.However, in this setting, the authors left open the problem of existence of sam-pling and interpolation sets with densities arbitrarily close to the critical one.In this talk we will show that indeed, sampling and interpolation sets near thecritical density do exist in the setting of LCA groups.

References

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[1] Grochenig K., Kutyniok G., Seip K., Landau’s necessary density condi-tions for LCA groups, J. Funct. Anal. 255 (2008) 1831-1850.

[2] Kohlenberg A., Exact interpolation of band-limited functions, J. Appl.Phys., 24, no. 12 (1935) 1432-1436,

[3] H. Landau, Necessary density conditions for sampling and interpolationof certain entire functions, Acta Math. 117 (1967) 37-52.

[4] Lyubarskii Y. I., Seip K., Sampling and intepolating sequences for multiband-limited Functions and exponential bases on disconnected sets, J. Fourier Anal.Appl., 3, no. 5 (1997) 597-615.

[5] B. Matei, Y. Meyer, Simple quasicrystals are sets of stable sampling,Complex Var. Elliptic Equ. 55 (2010), 947-964.

[6] Marzo J., Riesz basis of exponentials for a union of cubes in Rd, preprintarXiv: math/0601288.

Gabor systems with randomizationEnrico Au-YeungDepartment of Mathematics, University of British [email protected]

The short-time Fourier transform and the corresponding inversion formulaallow us to represent a function as a continuous superposition of time-frequencyatoms given by translations and modulations of a fixed window function. A goalof time-frequency analysis is to find a discrete expansion of a function analogousto the continuous version. Recently, several authors have investigated Gaborframes, where the points in the time-frequency plane are irregular; these pointsdo not form a lattice. Another direction of inquiry is to use several windowfunctions instead of fixing one window function. Along this line of research, weintroduce randomization to the Gabor system so that the window functions arerandom.

Finite extensions of Bessel sequencesDamir BakicUniversity of [email protected]

Coauthors: Tomislav Beric

We will characterize Bessel sequences in infinite-dimensional Hilbert spacesthat can be extended to frames by adding finitely many vectors. We will alsodiscuss finite-dimensional perturbations of Bessel sequences and some relatedtopics.

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Phase retrieval using Lipschitz continuous mapsRadu BalanUniversity of [email protected]

Coauthors: Dongmian Zou, University of Maryland

In this note we prove that reconstruction from magnitudes of frame coeffi-cients (the so called “phase retrieval problem”) can be performed using Lipschitzcontinuous maps. Specifically we show that when the nonlinear analysis mapα : H → Rm is injective, with (α(x))k = |〈x, fk〉|2, where f1, . . . , fm is aframe for the Hilbert space H, then there exists a left inverse map ω : Rm → Hthat is Lipschitz continuous. Additionally we obtain the Lipschitz constant ofthis inverse map in terms of the lower Lipschitz constant of α.

Noncommutative Zak transform and frame theoryDavide BarbieriUniversidad Autonoma de [email protected]

Coauthors: E. Hernndez, J. Parcet, V. Paternostro

Invariant subspaces with respect to several classes of unitary representationsof noncommutative discrete groups can be effectively studied in spaces of op-erators by means of a generalized Zak transform. This turns out to define asurjective isometry onto an operator valued Hilbert space satisfying a quasi-periodicity condition. Relevant examples that can be treated are left shifts ofcountable subgroups, which generalize Zn translations in L2(Rn) to e.g. thecorresponding situation in the Heisenberg group, or the quasiregular represen-tation of a semidirect product. The main proofs will be given together withconcrete examples.

Vector-valued ambiguity functions and balayageJohn J. BenedettoNorbert Wiener Center, University of [email protected]

We solve an algebraic and an analytic problem related through the STFT.For the algebraic problem, we define a vector-valued ambiguity function. The

motivation is modelling of multi-sensor environments. The technology involvesformulating and solving so-called frame multiplication problems in terms ofrepresentations of finite groups.

For the analytic problem, we use balayage to generalize Beurling’s and HenryLandau’s theory of Fourier frames to prove STFT and pseudo-differential oper-ator frame inqualities.

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The algebraic part is a collaboration with Travis Andrews and Jeff Donatelli.The analytic part is a collaboration with Enrico Au-Yeung.

Compression and signal processing using level-crossing samplingBrigitte Bidegaray-FesquetCNRS, Grenoble, [email protected]

An important issue in the design of mobile systems is to increase their auton-omy. One way to achieve this is to use asynchronous event-driven architectures.These systems take typically samples each time a level is crossed. This leadsto a reduced number of samples, compared to a Nyquist sampling but they arenon-uniform. Dedicated signal processing algorithms have to be developed todeal with these samples. Each single computation is usually more complex thanfor uniform signal processing chains, but this is largely compensated by thedrastic reduction of the number of samples. This form of signal compression isin particular studied for signals with known local Hlder regularity.

Dynamic equipartitioning of frame potentials: equiangular vs. equidis-tributed tight framesBernhard BodmannUniversity of [email protected]

Coauthors: John Haas

Equiangular tight frames are optimal designs for many applications. How-ever, their existence depends on dimensionality and size of the frame, and theredoes not seem to be a universal construction principle for all known exam-ples. In this talk we present dynamic optimization principles in frame designas an alternative to traditional group-theoretic or combinatorial constructionmethods. The feasibility of dynamic frame design is based on a result due toLojasiewicz which guarantees the convergence of the gradient descent on themanifold of Parseval frames for real analytic frame potentials. We introduceand discuss the class of equidistributed tight frames, which is more general thanequiangular tight frames and is shown to have an abundance of examples inany dimension. Finally, equidistributed tight frames are characterized as thecommon stationary points for a family of real analytic frame potentials.

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Toral eigenfuctions and their nodal setsJean BourgainInstitute for Advanced [email protected]

The spectral properties of the torus are in some sense the easiest to explorebecause the eigenfunctions are explicit trigonometric polynomials.Neverthelessseveral main conjectures on their distribution and the behavior of their zerosets are still far from resolved.These are often problems at the interface of har-monic analysis and number theory.In the talk,some of those aspects and recentadvances will be presented.We also briefly discuss the role of computer assistednumerics in the problem of nodal domain counting

A combinatorial characterization of tight fusion framesMarcin BownikUniversity of [email protected]

Coauthors: Kurt Luoto, Edward Richmond

In this talk we present a combinatorial characterization of tight fusion frame(TFF) sequences using Littlewood-Richardson skew tableaux. The equal rankcase has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Ourcharacterization does not have this limitation. We also develop some methodsfor generating TFF sequences. The basic technique is a majorization principlefor TFF sequences combined with spatial and Naimark dualities. We use thesemethods and our characterization to give necessary and sufficient conditionswhich are satisfied by the first three highest ranks. We exhibit four classes ofTFF sequences which have unique maximal elements with respect to majoriza-tion partial order. Finally, we give several examples illustrating our techniquesincluding an example of tight fusion frame which can not be constructed by theexisting spectral tetris techniques.

An algorithm for variable density sampling with block-constrainedacquisitionClaire BoyerInstitut de Mathmatiques de Toulouse (IMT), [email protected]

Coauthors: Pierre Weiss (ITAV, France), Jeremie Bigot (ISAE, France)

Reducing acquisition time is of fundamental importance in various imagingmodalities. The concept of variable density sampling provides an appealingframework to address this issue. It was justified recently from a theoreticalpoint of view in the compressed sensing (CS) literature. Unfortunately, the

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sampling schemes suggested by current CS theories may not be relevant sincethey do not take the acquisition constraints into account (for example, continu-ity of the acquisition trajectory in Magnetic Resonance Imaging - MRI). In thispaper, we propose a numerical method to perform variable density samplingwith block constraints. Our main contribution is to propose a new way to drawthe blocks in order to mimic CS strategies based on isolated measurements. Thebasic idea is to minimize a tailored dissimilarity measure between a probabil-ity distribution defined on the set of isolated measurements and a probabilitydistribution defined on a set of blocks of measurements. This problem turnsout to be convex and solvable in high dimension. Our second contribution isto define an efficient minimization algorithm based on Nesterov’s acceleratedgradient descent in metric spaces. We study carefully the choice of the metricsand of the prox function. We show that the optimal choice may depend on thetype of blocks under consideration. Finally, we show that we can obtain betterMRI reconstruction results using our sampling schemes than standard strategiessuch as equiangularly distributed radial lines.

Finding the right model for our dataCarlos CabrelliUniversity of Buenos Aires [email protected]

Finding the right model for given data Assume that we need to process alarge amount of data that we know is low dimensional. A common practice inthis case, is to impose some hypothesis on the data and use a standard model.For example we can assume that the data is band limited, then we can use anappropriate Paley-Wiener space. A more realistic approach is to select a bigclass of models (subspaces) and try to find the one that best fit our data. Inthis talk we will show that this is possible in many situations. In particular weillustrate this in the space L2(RN ), with shift invariant spaces, having extra-invariance, which provide a model for noisy data and give a way to control thejitter error.

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Robust and efficient compressed sensingJameson CahillDuke [email protected]

Coauthors: Peter Casazza and Dustin Mixon

In the design of measurement matrices for use in compressed sensing thereare two properties which have received a lot of attention, namely, the null spaceproperty (NSP) and the restricted isometry property (RIP). The RIP is usuallypreferred as it can be used to give good stability and robustness guarantees,and it is also known to imply the NSP. However, these two properties are fun-damentally different in that the NSP (as the name suggests) is only a propertyof the null space, whereas the RIP is an explicit property of the matrix itself.We show that, in fact, the RIP is in essence a property of the null space in thesense that having the same null space as a matrix with the RIP is just as goodas being a matrix with the RIP. We also so that in some situations it is betterto use a well conditioned matrix matrix with the same null space as a matrixwith the RIP.

L1 approximation via de Branges spaces and applicationsEmanuel CarneiroIMPA - Rio de [email protected]

Given a real-valued function, we address in this talk the problem of con-structing optimal one-sided approximations of prescribed exponential type tothis function. These approximations are optimal in the sense that they optimizea given weighted L1 metric over the real line. We shall see how this constructionis related to the theory of de Branges spaces of entire functions and commenta little on the interesting applications of these extremal functions, that include,for instance, new upper bounds for the pair correlation of zeros of the Riemannzeta-function.

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Phase retrieval by projectionsPeter G. CasazzaUniversity of [email protected]

Phaseless reconstruction has broad application to x-ray crystallography, elec-tron microscopy, diffractive imaging, x-ray tomography and more. The mathe-matics of phase retrieval is a very active area of research at this time. Recently,in several areas of research such as crystal twinning, it has become necessary todo phaseless reconstruction from the norms of the projections of a signal ontosubspaces. It was believed that norms of projections give much less informationthan inner products with vectors and so we would need many more projectionsto do phaseless than we can do with vectors. We will look at recent results onthis problem which include the surprising results that we can do phaseless re-construction with the same number of projections as we can do it with vectors.This now reverses the above problem to: Is it possible to do phaseless with fewerprojections than we can do it with vectors?

The analysis of periodic point processesStephen D. CaseyAmerican [email protected]

Our talk addresses the problems of extracting information from periodicpoint processes. These problems arise in numerous situations, from radar pulserepetition interval analysis to bit synchronization in communication systems.We divide our analysis into two cases - periodic processes created by a singlesource, and those processes created by several sources. We wish to extract thefundamental period of the generators, and, in the second case, to deinterleavethe processes.

We first present very efficient algorithm for extracting the fundamental pe-riod from a set of sparse and noisy observations of a single source periodicprocess. The procedure is computationally straightforward, stable with respectto noise and converges quickly. Its use is justified by a theorem which showsthat for a set of randomly chosen positive integers, the probability that they donot all share a common prime factor approaches one quickly as the cardinalityof the set increases. The proof of this theorem rests on a probabilistic inter-pretation of the Riemann zeta function. We then build upon this procedure todeinterleave and then analyze data from multiple periodic processes. This reliesboth on the the probabilistic interpretation of the Riemann zeta function, theequidistribution theorem of Weyl, and Wiener’s periodogram.

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Acceleration of nonlinear frame algorithm.Wenjian ChenSun Yat-sen University & University of Central [email protected]

Coauthors: Qiyu Sun

We develop a polynomial acceleration technique to improve Van-Cittert it-eration algorithm for solving localized nonlinear functional equations.

A null space property approach to compressed sensing with framesXuemei ChenUniversity of Maryland, College [email protected]

Coauthors: Haichao Wang Rongrong Wang

An interesting topic in compressive sensing concerns problems of sensing andrecovering signals with sparse representations in a dictionary. In this note, westudy conditions of sensing matrices A for the `1-synthesis method to accuratelyrecover sparse, or nearly sparse signals in a given dictionary D. In particular,we propose a dictionary based null space property (D-NSP) which, to the bestof our knowledge, is the first sufficient and necessary condition for the successof the `1 recovery. This new property is then utilized to detect some of thosedictionaries whose sparse families cannot be compressed universally. Moreover,when the dictionary is full spark, we show that AD being NSP, which is well-known to be only sufficient for stable recovery via `1-synthesis method, is indeednecessary as well. This is a joint work with Haichao Wang and Rongrong Wang.

Sampling in finite-dimensional reproducing kernel spacesCheng ChengUniversity of Central [email protected]

Coauthors: Yingchun Jiang and Qiyu Sun

In this poster, we will discuss sampling and reconstruction of signals in afinite-dimensional reproducing kernel space.

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The unitary extension principle on LCA groupsOle ChristensenTechnical University of [email protected]

Coauthors: Say Song Goh

We present an extension of the unitary extension principle by Ron & Shento the setting of LCA groups. In the general setting this yields a tight framefor L2 consisting of modulates of a collection of functions.

Hermite subdivision schemes and exponential polynomial generationMariantonia CotroneiUniversity of Reggio Calabria, [email protected]

Coauthors: Costanza Conti (University of Firenze, Italy), Tomas Sauer (Uni-versity of Passau, Germany)

Hermite subdivision schemes act on vector valued data interpreting theircomponents as function values and associated consecutive derivatives. In thistalk we address the problem of studying the exponential and polynomial preser-vation capability of such kind of schemes. The main tool for our investigationare convolution operators that annihilate the spaces generated by polynomialand exponential functions, which apparently is a general concept in the study ofvarious types of subdivision operators. Based on these annihilators, we charac-terize the so-called spectral condition in terms of factorization of the subdivisionoperator and we then show how this factorization can be used to examine theconvergence of the scheme.

Integrable wavelet transforms with abelian dilation groupsBrad CurreySaint Louis [email protected]

Coauthors: Hartmut Fuhr and Keith Taylor

We consider a class of semidirect products G = Rn · H, with H a suitablychosen abelian matrix group. The choice of H ensures that there is a waveletinversion formula. Motivated in part by coorbit theory, we are looking forcriteria to determine conditions under which there is a wavelet such that theassociated reproducing kernel is integrable.

It is well-known that the existence of integrable wavelet coefficients is relatedto the question whether the unitary dual contains open compact sets. Our maingeneral result reduces the latter problem to that of identifying compact opensets in the quotient space of all dual orbits of maximal dimension. This result is

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applied to study integrability for certain families of dilation groups; in particular,we give a complete characterization valid for connected abelian matrix groupsacting in dimension three.

Preimage Problem for Laplacian Eigenmaps with Applications toData Integration and RecoveryWojciech CzajaUniversity of Maryland College [email protected]

Coauthors: Alexander Cloninger and Timothy Doster

Nonlinear representation and data organization techniques became a vitalpart of machine learning and applied harmonic analysis, due to the their abilityto solve many problems related to complex, high-dimensional, large and noisydata sets. Many of these techniques are defined by means of appropriatelyrepresented operators on data graphs, e.g., graph Laplacian or Schroedinger op-erators. This analysis leads to representation of data in a new feature space,which is the target space of the nonlinear process. On the one hand, such spacemay be very useful for direct comparison of heterogeneous sensing modalities.On the other hand, however, its non-physical nature may cause difficulties inmany experimental applications. Our goal in this talk is to present an algorithmfor fast, approximate inversion of Laplacian Eigenmaps - a leading and populardimension reduction scheme. In our construction, we rely on Nystroem exten-sion principle, L1 regularization, and multidimensional scaling. Furthermore, weshall combine this new preimage methodology together with the feature spacerotations of Coifman and Hirn, to provide a unified methodology for data inte-gration and recovery. We shall illustrate the usefulness of our algorithms withexamples from remote sensing.

Frame properties of low autocorrelation stochastic waveformsSomantika DattaUniversity of Idaho, Department of Mathematics, Moscow, ID, [email protected]

Waveforms with spike like autocorrelation are desirable in waveform designand are particularly useful in areas of radar and communications. In this work,stochastic waveforms are constructed whose expected autocorrelation can bemade arbitrarily small outside the origin. Construction of stochastic framesfor finite-dimensional spaces from such waveforms is addressed and their frameproperties are studied using results from random matrix theory.

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Spatio-temporal sampling sets for stable recovery of signals in evolu-tionary systemsJacqueline DavisVanderbilt [email protected]

Coauthors: Akram Aldroubi, Ilya Krishtal

Dynamical sampling is a new class of sampling problems in which an evolvingsignal is sampled at various times. The spatio-temporal problem in dynamicalsampling asks the question: when do coarse spatial samplings of an evolvingsignal taken at varying times contain the same information as a finer spatialsampling taken at the earliest time? We show that under some conditions onthe evolving system it is possible to compensate for spatial undersampling bytaking additional time samples. In other words, spatial samples can be tradedfor time samples.

A Kantorovich Metric for Projection Valued MeasuresTrubee DavisonUniversity of [email protected]

Given a compact metric space X, the collection of Borel probability measureson X can be made into a complete metric space via the Kantorovich metric. Wegeneralize this well known result to projection valued measures. In particular,given a Hilbert space H, consider the collection of projection valued measuresfrom X into the projections onH. We show that this collection can be made intocomplete metric space via a generalized Kantorovich metric. As an application,we use the Contraction Mapping Theorem on this complete metric space ofprojection valued measures to provide an alternative method for proving a fixedpoint result due to P. Jorgensen. This fixed point, which is a projection valuedmeasure, arises from an iterated function system on X.

Structured measurements and ell0 recovery in arbitrarily coherentdictionariesLaurent [email protected]

Coauthors: Nam Nguyen

We present a very simple algorithm, called the superset method, for sparserecovery from linear measurements of the form Y = Adiag(x)B. Recovery isnot only possible in the regime of arbitrarily high dictionary coherence 1 - eps,but robust as long as the noise level is small in relation to eps. This type of

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behavior does not in general hold for ell1 minimization. The main applicationis to super-resolution from bandlimited samples, where A and B are partialFourier matrices.

Image restoration: variational models, PDEs and wavelet framesBin DongDepartment of Mathematics, Program in Applied Mathematics, University [email protected]

Coauthors: Jian-Feng Cai, Qingtang Jiang, Stanley Osher, Zuowei Shen

Mathematics has become one of the main driving forces of the modern devel-opment of image restoration. There are several mathematical approaches thathave been rather successful, including variational methods, partial differentialequations (PDEs) based methods, and wavelet frame based methods. This talkis based on two of our recent papers that established connections between thesepopular methods. Our key observation is that: wavelet frame transform canbe regarded as a discrete approximation of differential operators. However, thediscrete approximation by wavelet frames is fundamentally different from finitedifference approximate, due to their specific way of sampling derivatives. I willparticularly discuss: (1) how some wavelet frame based image restoration mod-els can be regarded as discrete approximation of a certain type of variationalmodel through Gamma-convergence; (2) how to design iterative wavelet frameshrinkage that solves various types of nonlinear evolution PDEs, especially thoseused for image restoration; (3) through our theoretically analysis, how can weobtain new insights to all of these approaches, and combine the merits of themto create new and better models solving image restoration problems or otherimportant and challenging problems in imaging science.

New Analysis of Manifold Embeddings and Signal Recovery fromCompressive MeasurementsArmin EftekhariColorado School of Mines / [email protected]

Coauthors: Michael B. Wakin

Compressive Sensing (CS) exploits the surprising fact that the informationcontained in a sparse signal can be preserved in a small number of compressive,often random linear measurements of that signal. Strong theoretical guaranteeshave been established concerning the embedding of a sparse signal family undera random measurement operator and on the accuracy to which sparse signals canbe recovered from noisy compressive measurements. In this work, we addresssimilar questions in the context of a different modeling framework. Instead of

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sparse models, we focus on the broad class of manifold models, which can arisein both parametric and non-parametric signal families. Using tools from thetheory of empirical processes, we improve upon previous results concerning theembedding of low-dimensional manifolds under random measurement operators.We also establish both deterministic and probabilistic instance-optimal boundsin `2 for manifold-based signal recovery and parameter estimation from noisycompressive measurements. In line with analogous results for sparsity-basedCS, we conclude that much stronger bounds are possible in the probabilisticsetting. Our work supports the growing evidence that manifold-based modelscan be used with high accuracy in compressive signal processing.

Sparse Signal Recovery From Nonlinear MeasurementsYonina EldarTechnion, [email protected]

We consider an extension of compressed sensing to nonlinear measurements.We present and analyze several different optimality criteria for sparse recoveryfrom nonlinear measurements which are based on the notions of stationarityand coordinatewise optimality and show that they can be used to develop effi-cient recovery algorithms. A special case that is of large interest in the area ofoptics is that of phase retrieval, in which one needs to recover an image givenonly its Fourier transform magnitude. We propose an efficient algorithm forphase retrieval and prove certain optimality properties of the proposed method.We also consider conditions on the number of measurements needed for stablephase retrieval and show that surprisingly the results coincide with those ob-tained in the linear measurement setting (up to constants). We demonstrateour algorithms and results on a variety of problems in optical imaging

Sparse approximations of spatially varying blur operators in the waveletdomainPaul EscandeDepartement Mathematiques, Informatique, Automatique (DMIA) Institut Superieurde l’Aeronautique et de l’Espace (ISAE) Toulouse, [email protected]

Coauthors: Jeremie Bigot (DMIA - ISAE - [email protected]), Pierre Weiss(ITAV - CNRS - [email protected])

Restoration of images blurred by spatially varying PSFs is a problem metincreasingly. One of the main difficulties is the computational burden caused bythe huge dimensions of blur matrices. It prevents the use of naive approachesto perform matrix-vector multiplications. We study an original approach whichconsists of approximating blurring operators by sparse matrices in the wavelet

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domain. We provide theoretical complexity results and compare this approachto standard approximations as piecewise convolutions. We finish by showingthat the sparsity pattern of the matrix can be pre-defined, which is central intasks such as blind deconvolution.

Keywords: Image deblurring, spatially varying blur, operator approxima-tions, wavelet transforms, sparse approximations, Calderon-Zygmund operators,structured sparsity.

Stable Recovery with Gauge RegularizationJalal FADILICNRS-ENSICAEN-Univ. [email protected]

Coauthors: Samuel Vaiter (CEREMADE CNRS-Univ. of Paris-Dauphine) GabrielPeyr (CEREMADE CNRS-Univ. of Paris-Dauphine) Mohammad Golbabaee(CEREMADE CNRS-Univ. of Paris-Dauphine)

Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than theambient dimension of the object to be estimated. A line of recent work has stud-ied regularization models with various types of low-dimensional structures. Insuch settings, the general approach is to solve a regularized optimization prob-lem, which combines a data fidelity term and some regularization penalty thatpromotes the assumed lowdimensional/ simple structure. This paper provides ageneral framework to capture this low-dimensional structure through what wecoin piecewise regular gauges. These are convex, non-negative, closed, boundedand positively homogenous functions that will promote objects living on low-dimensional subspaces. This class of regularizers encompasses many popularexamples such as the `1 norm, `1 − `2 norm (group sparsity), as well as severalothers including the `∞ norm. We will show that the set of piecewise regu-lar gauges is closed under addition and pre-composition by a linear operator,which allows to cover mixed regularization, and analysis-type priors (e.g. totalvariation, etc.). Our main results provide a unified sharp analysis of exact androbust recovery guarantees from partial measurements.

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Mitigating the data-deluge by an adequate sampling for low-powersystemsLaurent FesquetTIMA, Grenoble Institute of [email protected]

Coauthors: Tugdual Le Pelletier, Taha Beyrouthy, Brigitte Bidegaray-Fesquet

Today, our digital society exchanges data flows as never it has been the casein the past. The amount of data is incredibly large and the future promises thatnot only human will exchange digital data but also technological equipment,robots, etc. We are close to open the door of the internet of things. This dataorgy wastes a lot of energy and contributes to a non-ecological approach ofour digital life. Indeed, Internet and the new technologies consume about 10Italready exists design solutions to enhance the energetic performances of theelectronic systems and circuits, of the computers and their mobile applications:a lot of techniques and also a lot of publications! Nevertheless, another wayto reduce energy is to rethink the sampling techniques and digital processingchains. Considering that our digital life is dictated by the Shannon theory, weproduce more digital data than expected, more than necessary. Indeed, uselessdata produce more computation, more storage, more communications and alsomore power consumption. If we disregard the Shannon theory, we can discovernew sampling and processing techniques. A small set of ideas is given throughexamples such as filters, pattern recognition techniques, etc. but the PandorasBox has to be opened to drastically reduce the useless data and mathematiciansprobably have a key role to play in this revolution.

This study is supported by the Persyval-lab project.

Constructing equi-isoclinic tight fusion framesMatthew FickusAir Force Institute of [email protected]

Coauthors: John Jasper, Dustin G. Mixon, Jesse D. Peterson

A collection of equi-dimensional subspaces is an equi-isoclinic tight fusionframe (EITFF) for a given Euclidean space if it is a tight fusion frame for thatspace and if the principal angles between any pair of subspaces are all the same.When they exist, EITFFs are optimal packings in the Grassmannian. They areideal for block-coherence-based compressed sensing, permitting reconstructionof sufficiently sparse signals via, for example, block orthogonal matching pur-suit. The easiest way to construct an EITFF is to take a tensor product of anequiangular tight frame with an orthonormal basis. We discuss some new non-trivial generalizations of this construction, including one method that involvesfilter banks and another which uses harmonic frames.

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Time-frequency decompositions in seismic data analysisSergey FomelThe University of Texas at [email protected]

Seismic wave propagation in the Earth can be strongly affected by frequencyattenuation. In addition, seismic signals exhibit variations in local slope, whichcan be described using non-stationary spectral analysis. For these reasons, time-frequency decompositions play an important role in seismic data processing,where they are used for signal enhancement and parameter estimation. I willdescribe both recent geophysical applications of time-frequency decompositionsand recently developed time-frequency analysis techniques, such as regularizednonstationary regression.

Interpolation with complex B-splinesBrigitte ForsterUniversity of Passau, [email protected]

Coauthors: Ramunas Garunkstis (Vilnius University, Lithuania), Peter Masso-pust (Helmholtz Zentrum Munchen, Germany) and Jorn Steuding (WurzburgUniversity, Germany)

Cardinal B-splines of complex order or, for short, complex B-splines, are nat-ural extensions of classical Curry-Schoenberg (polynomial) B-splines Bn, wherethe order n ∈ N is replaced by a complex number s. These complex B-splinesinherit many of the important and interesting properties of the Bn. In this talk,we will concentrate on the interpolation property. Whereas for the fractionalB-splines Bα, α > 1, the interpolation property can be easily verified, this isnot obvious for the complex case. In fact, this question is closely related to thegrowth conditions and the distribution of zeros of sums of Hurwitz zeta func-tions. In the talk, we give a positive answer to the question of interpolationwith complex B-splines for a certain range of complex degrees. The completecharacterization of the admissible degrees is still an open question.

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Exponentially decaying reconstruction error in one-bit compressivesensingSimon FoucartUniversity of [email protected]

Coauthors: R. Baraniuk, D. Needell, Y. Plan, and M. Wootters

In one-bit compressive sensing, s-sparse vectors x ∈ Rn are acquired viaextremely quantized linear measurements yi = sgn〈ai,x〉, i = 1, . . . ,m. Sev-eral procedures to reconstruct these sparse vectors have been shown to beeffective when a1, . . . ,am are independent random vectors. They typicallyyield an error decaying polynomially in λ := m/(s log(n/s)). This rate can-not be improved in the measurement framework described above. However,we show that a reconstruction error decaying exponentially in λ is achievablewhen thresholds τ1, . . . , τm are chosen adaptively in the quantized measure-ments yi = sgn(〈ai,x〉 − τi), i = 1, . . . ,m. Our procedure, which is robustto measurement error, is based on a simple recursive scheme involving eitherhard-thresholding or linear programming.

Coordinate systems of translates of a single function in Lp(R)Daniel FreemanSt Louis [email protected]

Coauthors: Edward Odell (University of Texas), Thomas Schlumprecht (TexasA&M University), Andras Zsak (Peterhouse College, Cambridge University)

Wavelet coordinate systems are formed by starting with a single function inL2 which is then translated and dilated to form a basis or a frame for L2. Weinvestigate what coordinate systems can be formed of just translations of a singlefunction. Christensen, Deng, and Heil proved that a sequence of translationsof a single function in L2 cannot form a frame for L2. In contrast to this, weshow that for all p > 2, Lp has an unconditional Schauder frame formed by theinteger translates of a single function in Lp. Furthermore, for all 1 ≤ p, Lp doesnot have an unconditional basis formed by translations of a single function inLp.

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Multiresolution Equivalence and Path-ConnectednessVeronika FurstFort Lewis Collegefurst [email protected]

Coauthors: Erich McAlister

An equivalence relation between multiresolution analyses was first intro-duced in 1996; an analogous definition for generalized multiresolution analyseswas given in 2010. Both types of equivalence classes are path-connected in anoperator-theoretic sense. Moreover, whenever two MRAs in L2(R) are equiva-lent, the GMRA path construction between their corresponding canonical GM-RAs yields the natural analog of the MRA path.

Frames of multi-windowed exponentials on subsets of RdJean-Pierre GabardoMcMaster [email protected]

Coauthors: Chun-Kit Lai

Given discrete subsets Λj ⊂ Rd, j = 1, . . . , q, consider the set of windowedexponentials

⋃qj=1gj(x)e2πiλ·x : λ ∈ Λj on L2(Ω). We show that a neces-

sary and sufficient condition for the windows gj to form a frame of windowedexponentials for L2(Ω) with some set of frequencies Λj , j = 1, . . . , q, is thatm ≤ maxj∈J |gj | ≤M almost everywhere on Ω for some subset J of 1, · · · , qand some positive constants m,M . If Ω is unbounded, we show that there is noframe of windowed exponentials if the Lebesgue measure of Ω is infinite. If Ω isunbounded but of finite measure, we give a sufficient condition for the existenceof Fourier frames on L2(Ω). At the same time, we also construct examples ofunbounded sets with finite measure that have no tight exponential frame.

Nonuniform generalized sampling and stable recovery of multivariatesignalsMilana GataricUniversity of [email protected]

Coauthors: Ben Adcock (Purdue University) Anders Hansen (University ofCambridge)

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In this talk, we deal with the problem of multivariate signal recovery froma finite collection of pointwise samples of its Fourier transform taken nonuni-formly. This problem is a major research topic, in both pure and applied math-ematics, driven by numerous practical applications ranging from Magnetic Res-onance Imaging (MRI) to Computed Tomography (CT), seismology and mi-croscopy, where the samples are fixed and acquired on sampling schemes notnecessarily Cartesian. We show that the desired reconstruction can be carriedout stably, accurately and efficiently in any given finite-dimensional approxima-tion subspace, within the so-called nonuniform generalized sampling framework.This is feasible under certain sufficient conditions on the sampling points —specifically, on their density and bandwidth — which are allowed to be highlynonuniform, i.e. the separation condition is not required. In particular, thesufficient density condition we provide is sharp and does not depend on the di-mension. This density characterization of a sampling scheme fills an importantgap in the existing nonuniform sampling theory literature, and it is based onour novel results on weighted Fourier frames which are improvement of existingresults of Grochenig and Beurling. Moreover, in one dimension, we analyze theimportant case when the approximation space consists of compactly supportedwavelets. We show that a linear scaling of the dimension of the space with thesampling bandwidth is both sufficient and necessary for stable recovery. Hence,up to constant factors, wavelets provide optimal bases for reconstruction.

Asymptotical analysis of inpainting via universal shearlet systems andclustered sparsityMartin GenzelTechnische Universitt [email protected]

Coauthors: Gitta Kutyniok (Technische Universitt Berlin)

Inpainting is generally understood to be the process of recovering corrupteddata signals. Problems with damaged or missing data often occur in the field ofimaging science; just imagine, for example, seismic data measured by an arrayof sensors with a more or less large number of them failing due to electronicdefects. This will result in an image with a bunch of small missing stripes. Inthe past decades, various algorithms have been developed to solve the problem of(image-)inpainting. However, most of the respective approaches are of empiricalnature, and a theoretical foundation is often missing.

To meet this challenge, the novel concept of clustered sparsity has beendeveloped recently and turned out to be a useful tool in the mathematicalanalysis of inpainting. The fundamental idea underlying this method is based onthe assumption that a given signal in Hilbert space H possesses some “significantstructure”, which can be “efficiently” represented by an appropriate Parsevalframe Φ for H.

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In this presentation, an abstract inpainting framework is introduced whichprovides recovering algorithms as well as some suitable error estimates. Basedon clustered sparsity, they are motivated by the ideas of compressed sensing.Concerning two dimensional images, these methods will be applied to a contin-uous model in L2(R2), which is governed by a line distribution. To capture thisanisotropic feature, we will choose Φ to be an universal shearlet system. Thisnovel construction of smooth band-limited Parseval frames particularly enablesa uniform treatment of both shearlets and wavelets. Provided that the gaps ofthe corrupted image are appropriately bounded, the main result finally showsthat one can indeed achieve asymptotically perfect inpainting. In this context,it turns out that, compared to classical wavelets, shearlets require much lowerassumptions on the gap size.

AN ADAPTIVE MESHFREE SPECTRAL GRAPH WAVELET METHODFOR PARTIAL DIFFERENTIAL EQUATIONS ON THE SPHEREKavita GoyalIndian Institute of Technology [email protected]

Coauthors: Mani Mehra

This paper proposes an adaptive meshfree spectral graph wavelet method tosolve partial differential equations on the sphere. The method uses multiresolu-tion analysis based on spectral graph wavelet for adaptivity. It uses radial basisfunctions for interpolation of functions and for approximation of the differentialoperators. The set of scattered node points is subject to dynamic changes atrun time which leads to adaptivity. The beauty of the method lies in the factthat the same operator is used for the approximation of differential operatorsand for the construction of spectral graph wavelet. Initially, we have appliedthe method on spherical diffusion equation. The problem of pattern formationon the surface of the sphere (using Turing equations) is addressed to test thestrength of the method. The numerical results show that the method can accu-rately capture the emergence of the localized patterns at all the scales and thenode arrangement is accordingly adapted. The convergence of the method isverified. For each test problem, the CPU time taken by the proposed method iscompared with the CPU time taken by a traditional method (spectral methodusing radial basis functions). It is observed that the adaptive meshfree spectralgraph wavelet method is highly efficient.

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The mystery of Gabor framesKarlheinz GrochenigUniversity of Vienna, [email protected]

Gabor frames remain mysterious objects in time-frequency analysis. Wewill first review some structural results about Gabor frames over a lattice; thisis the coarse structure of Gabor frames. For the fine structure one needs tounderstand which lattices generate a Gabor frame for a given window. Thefine structure of Gabor frames is largely unknown, there are few results andmany open conjectures. We will mention some recent results and formulatesome explicit conjectures. .

Matrix Fourier multipliers for Parseval multi-wavelet framesDeguang HanUniversity of Central [email protected]

Matrix Fourier multipliers are matrices with L∞-function entries that mapParseval multi-wavelet frames to Parseval multi-wavelet frames. Like Fourierwavelet multiplier, matrix Fourier multipliers can be used to derive new multi-wavelet frames and can help us better understand the basic theory on multi-wavelet frames. In this talk I will discuss a characterization of such matrixFourier multipliers

Compressed sensing and analog inverse problems - the need for a newtheoryAnders HansenUniversity of [email protected]

Coauthors: Ben Adcock, Clarice Poon, Bogdan Roman

Compressed sensing is based on the three pillars: sparsity, incoherence anduniform random subsampling. In addition, the concepts of uniform recoveryand the Restricted Isometry Property (RIP) have had a great impact. Intrigu-ingly, in an overwhelming number of analog inverse problems where compressedsensing is used or can be used (such as MRI, X-ray tomography, Electron mi-croscopy, Reflection seismology etc.) these pillars are absent. Moreover, easynumerical tests reveal that with the successful sampling strategies used in prac-tice one does not observe uniform recovery nor the RIP. In particular, none ofthe existing theory can explain the success of compressed sensing in a vast areawhere it is used. In this talk we will demonstrate how real world analog inverse

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problems are not sparse, yet asymptotically sparse, coherent, yet asymptoti-cally incoherent, and moreover, that uniform random subsampling yields highlysuboptimal results. In addition, we will present easy arguments explaining whyuniform recovery and the RIP is not observed in practice. Finally, we will in-troduce a new theory that aligns with the actual implementation of compressedsensing that is used in applications. This theory is based on asymptotic spar-sity, asymptotic incoherence and random sampling with different densities. Thistheory supports two intriguing phenomena observed in reality: 1. the success ofcompressed sensing is resolution dependent, 2. the optimal sampling strategyis signal structure dependent. The last point opens up for a whole new area ofresearch, namely the quest for the optimal sampling strategies.

Different faces of the shearlet transformSoren HauserTU Kaiserslautern, [email protected]

Coauthors: Stephan Dahlke (Philipps-Universitat Marburg, Germany), Filippode Mari (Universita degli Studi di Genova, Italy), Ernesto de Vito (Universitadegli Studi di Genova, Italy), Gabriele Steidl (TU Kaiserslautern, Germany),Gerd Teschke (Hochschule Neubrandenburg, Germany)

Recently, shearlet groups have received much attention in connection withshearlet transforms applied for orientation sensitive image analysis and restora-tion. The square integrable representations of the shearlet groups provide notonly the basis for the shearlet transforms but also for a very natural definitionof scales of smoothness spaces, called shearlet coorbit spaces. The aim of thistalk is twofold: first we discover isomorphisms between shearlet groups andother well-known groups, namely extended Heisenberg groups and subgroups ofthe symplectic group. Interestingly, the connected shearlet group with positivedilations has an isomorphic symplectic subgroup, while this is not true for thefull shearlet group with all nonzero dilations.

Having understood the various group isomorphisms it is natural to ask forthe relations between coorbit spaces of isomorphic groups with equivalent rep-resentations. These connections are examined in the second part of the talk.We describe how isomorphic groups with equivalent representations lead to iso-morphic coorbit spaces. In particular we apply this result to square integrablerepresentations of the connected shearlet groups and metaplectic representationsof subgroups of the symplectic group. This implies the definition of metaplecticcoorbit spaces.

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Linear Independence of Time-Frequency TranslatesChristopher HeilGeorgia [email protected]

The Linear Independence of Time-Frequency Translates Conjecture, alsoknown as the HRT conjecture, states that any finite set of time-frequency trans-lates of a given L2 function must be linearly independent. This conjecture, whichwas first stated in print in 1996, remains open today. We will discuss this con-jecture, its context, and the (frustratingly few) partial results that are currentlyavailable.

Riesz and Frame sequences generated by unitary actions of discretegroupsEugenio HernndezUniversidad Autnoma de [email protected]

Coauthors: Davide Barbieri and Javier Parcet

We characterize Riesz and frame sequencies which arise from the action of acountable discrete group Γ on a single element of a given Hilbert space H. As Γmight not be abelian, this is done in terms of an operator-valued bracket maptaking values in the L1-space associated to the group von Neumann algebraof Γ. Our result generalizes recent work for locally compact abelian groups ofHernandez, Sikic, Weiss and Wilson. In many cases, the bracket map can bedecribed in terms of a noncommutative form of the Zak transform.

Minimal C1,1 extensionsMatthew HirnEcole normale [email protected]

Coauthors: Ariel Herbert-Voss, Erwan Le Gruyer, Frederick McCollum

Consider the following Whitney type extension problem for C1,1(Rd). Sup-pose we are given a closed subdomain E of Rd, along with a 1st degree polyno-mial at each point in E that specifies a potential function value and a gradient.We refer to such information as a ”1-field.” Some questions one might ask arethe following: When can we extend the 1-field to a function F in C1,1(Rd) suchthat F agrees with the specified function values and gradients on E? If such anF exists, how small can we make the Lipschitz constant of the gradient of F?The first question was answered by Whitney in 1934 as part of the more generalWhitney Extension Theorem. The second question, however, was only recentlyanswered in 2009 by Le Gruyer. In this talk we will discuss these results, and

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then explore some of the ramifications of them. In particular, we will generalizethe concept of an absolutely minimal Lipschitz extension (AMLE) to 1-fieldsand more general extension problems, and prove the existence of quasi-AMLEsin this expanded framework. Additionally, we present a practical, efficient algo-rithm for computing AMLEs of 1-fields when the subdomain E is finite. Thistalk is based on joint work with Ariel Herbert-Voss, Erwan Le Gruyer, andFrederick McCollum.

Clifford-Fourier transforms and waveletsJeff HoganUniversity of Newcastle, [email protected]

Coauthors: Andrew Morris (University of Newcastle, Australia)

We present some recent results in Clifford analysis aimed at the treatmentof multi-channel signals. A covariant Clifford-Fourier transform is introducedand its kernel computed in even dimensions. A convolution theorem will beproved, and applications to the construction of Clifford-valued wavelets andmultiwavelets explored.

Frame-based multi-scale Gaussian beams and parametrix construc-tion for wave equationsMaarten V. de HoopPurdue [email protected]

Coauthors: Michele Berra and Jos-Luis Romero

We construct frame-based multi-scale Gaussian beams following the dyadicparabolic decomposition of phase space. Using these, we generate a parametrixfor wave equations. Our parametrix is the sum of purely Gaussian functions andsolves Cauchy initial value problems. We establish an error estimate in terms ofthe scale content of the initial data. We then proceed with a construction of thecorresponding approximate solution of the inhomogeneous wave equation with aboundary source, converging in the limit of fine scales also. This solution playsa role in many applications in electrical engineering and reflection seismology.We mention the connection with wave atoms, and related work by Laptev andSigal (2000) and Bao et al. (2013).

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Optimal N-term approximation by linear splines on triangulationsArmin IskeUniversity of Hamburg, [email protected]

Coauthors: Laurent Demaret

Anisotropic triangulations provide efficient geometrical methods for sparserepresentations of bivariate functions from discrete data, in particular from im-age data. In previous work, we have proposed a locally adaptive method forefficient image approximation, called adaptive thinning, which relies on linearsplines over anisotropic Delaunay triangulations. In this talk, we discuss asymp-totically optimal N -term approximation rates for linear splines over anisotropictriangulations, where our analysis applies to three relevant classes of targetfunctions: (a) piecewise linear horizon functions across Holder smooth bound-aries, (b) functions of Wα,p regularity, where α > 2/p− 1, (c) piecewise regularhorizon functions of Wα,2 regularity, where α > 1.

Fast computation of n-widths via greedy least-squaresMark IwenMichigan State [email protected]

Coauthors: Felix Krahmer

I will discuss fast and deterministic dimensionality reduction techniques fora family of subspace approximation problems. Let S ⊂ RN be a given set ofM points. The techniques discussed find an O(n logM)-dimensional subspacethat is guaranteed to always contain a near-best fit n-dimensional hyperplane Hfor S with respect to the cumulative projection error

(∑x∈S ‖x−ΠHx‖p2

)1/p,

for any chosen p > 2. The deterministic algorithm runs in O(MN2

)-time,

and can be randomized to run in only O (MNn)-time while maintaining itserror guarantees with high probability. In the case p = ∞ the dimensionalityreduction techniques can be combined with efficient algorithms for computingthe John ellipsoid of a data set in order to produce an n-dimensional subspacewhose maximum `2-distance to any point in the convex hull of P is minimized.The resulting algorithm remains O (MNn)-time. These methods allow the fastcomputation of tightly fitting bounding regions in large and high-dimensionaldata sets for use in, e.g., database indexing schemes.

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When Buffon’s needle problem helps in quantizing the Johnson-LindenstraussLemmaLaurent JacquesISPGroup, ICTEAM/ELEN, University of Louvain (UCL), [email protected]

In 1733, Georges-Louis Leclerc, Comte de Buffon in France, set the groundof geometric probability theory by defining an enlightening problem: What isthe probability that a needle thrown randomly on a ground made of equispacedparallel strips lies on two of them?

In this presentation, we will show that the solution to this problem, and itsgeneralization to N dimensions, allows us to discover a quantized form of theJohnson-Lindenstrauss (JL) Lemma, i.e., one that combines a linear dimension-ality reduction procedure with a uniform quantization of precision δ > 0.

In particular, given a finite set S ⊂ RN of S points and a distortion levelε > 0, as soon as M > M0 = O(ε−2 logS), we can (randomly) construct amapping from (S, `2) to ((δ Z)M , `1) that approximately preserves the pairwisedistances between the points of S.

Interestingly, compared to the common JL Lemma, the mapping is quasi-isometric and we observe both an additive and a multiplicative distortions on theembedded distances. These two distortions, however, decay as O((logS/M)1/2)when M increases. Moreover, for coarse quantization, i.e., for high δ comparedto the set radius, the distortion is mainly additive, while for small δ the embed-ding tends to a Lipschitz isometric embedding.

We will also explain that there exists “almost” a quasi-isometric embeddingof (S, `2) in ((δZ)M , `2). This one involves a non-linear distortion of the `2-distance in S that vanishes for distant points in this set. Noticeably, the additivedistortion in this case is slower and decays as O((logS/M)1/4).

Finally, the presentation will be illustrated by simple numerical simulationsshowing that the additive and the multiplicative errors behave as predicted whenδ and M vary.

Wavelet techniques for p-exponent multifractal analysisStephane JaffardUniversity Paris [email protected]

Coauthors: Patrice Abry, Roberto Leonarduzzi, Clotilde Melot, Stephane Roux,Maria Eugenia Torres, Herwig Wendt

The purpose of multifractal analysis is the construction of scaling functionsthat allow to estimate the fractal dimensions of the pointwise singularities ofa signal. Independently of this theoretical interpretation, scaling functions arealso used as an efficient tool for model selection and parameter estimation.However, up to now, this technique could only be applied for data which arelocally bounded, an a priori hypothesis which is seldom met by real-life signals.

28

We will present an alternative of this method, which is obtained when theusual pointwise Hlder regularity is replaced by the p-exponent (a notion intro-duced by Calderon and Zygmund in the 1960s, in the context of PDEs). Thea priori assumption now is that the data locally belong to Lp, an assumptionwhich is much more frequently met by experimental data. We will presentthe mathematical background of these developments, and in particular how toderive the relevant wavelet-based scaling functions. We will also show applica-tions to stochastic processes and real-life data (signals and images), for whichthe previous approach based on the Holder exponent did not work.

Characterization of wavelets and MRA wavelets on local fields ofpositive characteristicQaiser JahanIndian Statistical Institute Kolkata, [email protected]

Coauthors: Biswaranjan Behera (Indian Statistical Institute Kolkata, India)

We provide a characterization of wavelets on local fields of positive charac-teristic based on results on affine and quasi affine frames. This result generalizesthe characterization of wavelets on Euclidean spaces by means of two basic equa-tions. We also give another characterization of wavelets. Further, all waveletswhich are associated with a multiresolution analysis on such a local field arealso characterized.

Correspondence between Wavelet Frame Shrinkage and NonlinearDiffusionQingtang JiangUniversity of Missouri - St. [email protected]

Coauthors: Bin Dong and Zuowei Shen

In this talk we will discuss the correspondence between the wavelet frameshrinkage and nonlinear diffusion. The connection will provide new and inspiringinterpretations of these both approaches for image restoration. In particular, thewavelet frame shrinkage algorithms that are commonly used in image restora-tion, such as the iterative soft-thresholding algorithms, lead to new types ofnonlinear PDEs that have not been considered in the literature. On the otherhand, the nonlinear PDE based approach also provides new insights into thedesirable choices of adaptive thresholds for wavelet frame shrinkage and enableus to design better wavelet frame algorithms for image restoration.

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Cardinal splines in piecewise constant tensionMasaru KamadaIbaraki [email protected]

The cubic spline gives the smoothest interpolation of data in the sense that itminimizes the integral of its squared second derivative. The linear spline givesthe shortest polygonal interpolation. The spline in tension was devised as ageneralization of those two splines. It minimizes the integral of a weighted sumof the squared first and second derivatives. The weight on the first derivativeis called tension because its increase makes the spline in tension approach theshortest linear spline while retaining smoothness like the cubic spline. Thetension has been fixed as a single constant over the entire domain since its firstappearance in 1966.

In this talk, we shall look at the spline in tension as the output of a lineardynamical system model with a series of delta functions input. In exchange ofrestricting the sampling points to be uniform, we can place a different tensionin each sampling interval because the control theory allows for time-varyingdynamics. Solving the dead beat control problem of the system, we will eventu-ally have a locally supported basis for the space of cardinal splines in piecewiseconstant tension.

An application is an adaptive image interpolation algorithm made up of theunivariate spline interpolations in the horizontal and vertical directions. Varyingthe tension in proportion to an index of sharp change in brightness, we obtainimage interpolation results with less ringing artifacts compared to those by thecubic spline interpolation.

We will also discuss a similar model for bivariate splines in piecewise constanttension which may generalize the bicubic spline interpolation.

Total Activation: Generalized Total Variation Regularization for Func-tional Magnetic Resonance Imaging Data AnalysisF. Isik KarahanogluEcole Polytechnique Federale de Lausanne, University of [email protected]

Coauthors: Dimitri Van De Ville

Total Variation (TV) regularization is an edge-preserving method extensivelyused in image and signal processing for denoising, restoration, and deconvolu-tion. TV regularization implicitly assumes a generating system that consistsof piecewise constant signals. We extend the notion of TV in the sense thatthe underlying generating system is represented as the combination of weightedand shifted Green’s function of a general differential operator, L, instead of thefirst order differential operator, D, in TV regularization. We cast a denoisingproblem whose regularization term combines sparsity constraint and discrete

30

differential filter associated to the inverse of the underlying system; i.e., ”anal-ysis” formulation. Further tailoring the operator enables to handle differentdriving signals; i.e., n-th order polynomials.

Employing 1-D generalized TV, we develop a novel spatio-tempoal deconvo-lution scheme, for which we coin the term Total Activation (TA), for functionalmagnetic resonance imaging (fMRI) data analysis. Specifically, we tune the gen-eral differential operator, L, to invert the transfer function of fMRI system; i.e.,hemodynamic response function. TA aims to recover the underlying activity-inducing signals, which are closely related to neuronal activity, without anyrestrictions on the timing or duration of the activations. That allows for detec-tion of spontaneous brain activity and observation of the non-stationary dynam-ics. The variational formulation is convex and contains a data-fitting term andtwo regularizers for the different dimensions of the data. First, temporal regu-larization identifies the ”innovation” signal (which is spike-type) as the sparsedriver of the hemodynamic system. Spatial regularization is incorporated usingmixed-norm based on anatomical priors of brain regions; i.e., activities in thesame brain regions are favored to be coherent. We employ the efficient gener-alized forward-backward splitting algorithm, which is a fast iterative shrinkagealgorithm that alternates between temporal and spatial domain solutions un-til convergence to the final estimate of the underlying activity-inducing signalis reached. We show that TA is able to recover the underlying activations infMRI experiment using visual stimuli. Moreover, we present that TA disentan-gles meaningful brain networks driven by short transitions during resting-statefMRI.

Prolate spheroidal wave functions: spectral analysis of the associateddifferential and sinc kernel operators with some related applicationsAbderrazek KarouiUniversity of Carthage, Faculty of Sciences of Bizerte, [email protected]

Coauthors: Aline Bonami, MAPMO, University of Orleans, France.

For fixed real number c > 0, the prolate spheroidal wave functions (PSWFs)(ψn,c)n≥0 form a basis with remarkable properties for the space of band-limitedfunctions with bandwidth c. They have been largely studied and used in var-ious classical signal processing applications as well as in some emerging andpromising subjects and area such as the performance of MIMO transmissionsin wireless network, random matrices and compressed sensing.

Note that the PSWFs were first known as the bounded eigenfunctions ofthe Sturm-Liouville operator’s Lc defined on C2([−1, 1]) by Lc(ψ) = −(1 −x2) d

2ψdx2 + 2x dψd x + c2x2ψ. A breakthrough in the subject of the PSWFs is due to

D. Slepian, H. Landau and H. Pollack, who have shown that the ψn,c are also

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the eigenfunctions of the integral operators Fc and Qc = c2πF

∗c Fc, defined on

L2([−1, 1]) by

Fc(f)(x) =

∫ 1

−1

ei c x yf(y) dy, Qc(ψ)(x) =

∫ 1

−1

sin c(x− y)

π(x− y)ψ(y) dy. (1)

As a result, the PSWFs exhibit the unique properties to form an orthogonal ba-sis of L2([−1, 1]), and an orthonormal basis of Bc = f ∈ L2(R), Support f ⊂[−c, c], the Paley-Wiener space of c−band-limited functions. We let (λn(c))ndenotes the infinite sequence of the eigenvalues of Qc, arranged in the decreasingorder. Most of the above mentioned applications of the PSWFs heavily rely ontheir explicit analytic properties as well as on the precise behaviour and decayrate of the corresponding eigenvalues (λn(c))n≥0. Although, there exists a richliterature on the numerical computation and asymptotic behaviours of the ψn,cand λn(c), very little is know about their explicit estimates.

The main purpose of this talk is to give a description of the recent results wehave recently obtained regarding the problem of the accurate explicit estimatesof the PSWFs and their associated eigenvalues. To this end, we first give somenew and useful bounds of the ψn,c and the eigenvalues χn(c) of the differentialoperator Lc. Then, under the condition that q = c2/χn(c) < 1, we prove thatψn,c is uniformly approximated on [−1, 1] by a single function involving thefunction J0 : the Bessel function of first kind and order zero. A special interest isdevoted to the computation of normalisation constant appearing in this uniformapproximation of the ψn,c so that the L2([−1, 1])−norm of this later equals 1. Asan important consequence of this uniform approximation, we prove that underthe condition q = c2/χn(c) < 1, the exact term of the exponential decay rate ofthe λn(c) is given by

λn(c) =1

2exp

(−π

2

2n

∫ 1

Φ( 2cπn )

1

t(E(t))2dt

). (2)

Here, E(x) =

∫ 1

0

√1− x2t2

1− t2dt is the elliptic Legendre integral of the second

kind and Φ = Ψ−1, where Ψ(x) =x

E(x), 0 ≤ x ≤ 1. Also, we give the quality of

approximation by the PSWFs in the spaces of band-limited, almost band-limitedfunctions as well as on the Sobolev spaces Hs([−1, 1]), s > 0. The differentresults of this talk, are illustrated by some numerical examples.

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Sparse approximations with α-moleculesSandra KeiperTU [email protected]

Coauthors: Philipp Grohs, Gitta Kutyniok, Martin Schfer

Cartoon-like image are a good known model to describe natural images.They are typically defined to be functions in R2 which are Holder continuousonto two areas separated by a α-Holder continuous boundary curve. One canshow that the error of the N -term approximation for such images cannot exceedN−α. Also well known is that shearlets and curvelets are well suited for suchanisotropic structures. In a recent paper we have shown that we can abstractmany anisotropic representation systems, like shearlets but also wavelets, to thenotion of α-molecules. This allows us to deduce general results like optimallysparse approximations. So we have shown that the shearlet transform associatedto the coefficient α as well as the curvelet transform provide almost optimallysparse approximations. Since shearlets are more suitable for the digital realmwe use shearlets for implementation.

Wiener algebra and Gabor frames with infinite supportYeonhyang KimCentral Michigan [email protected]

A frame in a Hilbert spaceH is a sequence of vectors (fi) for which there existconstants 0 < A ≤ B < ∞; such that for all f in H, A||f ||2 ≤

∑|〈f, fi〉|2 ≤

B||f ||2. Due to some desirable properties, frames with infinite support are ofinterest in applications and have been extensively studied. In this paper, weexplore the question of when we can generate a frame with infinite support forL2-space using frame techniques and the Wiener algebra.

Self-similar operator on a Bernoulli convolutionKeri KornelsonUniversity of [email protected]

Coauthors: Palle Jorgensen, Karen Shuman

Some, but not all, Bernoulli convolution measures are known to be spectral,in the sense that there exists a Fourier basis for the corresponding L2 Hilbertspace. The support of the Bernoulli convolutions are self-similar fractal subsetsof the real line. We describe an operator mapping between Fourier bases on onesuch spectral measure and find that it, too contains self-similarity properties.In this talk, we will describe the properties of this “operator-fractal”.

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Factorization results in matrix Banach algebrasIlya KrishtalNorthern Illinois [email protected]

Coauthors: Thomas Strohmer, Tim Wertz

We present conditions for a matrix in a Banach algebra that admits an LU -factorization in B(`2) to admit an LU -factorization in the Banach algebra. Afew other classical factorizations follow.

α-Molecules: Wavelets, Shearlets, and beyondGitta KutyniokTechnische Universitt [email protected]

Coauthors: Philipp Grohs (ETH Zurich), Sandra Keiper (Technische Univer-sitat Berlin), and Martin Schafer (Technische Universitat Berlin)

The area of applied harmonic analysis provides a variety of multiscale sys-tems such as wavelets, curvelets, shearlets, or ridgelets. A distinct propertyof each of those systems is the fact that it sparsely approximates a particu-lar class of functions. Some of these systems even share similar approximationproperties such as curvelets and shearlets which both optimally sparsely approx-imate functions governed by curvilinear features, a fact that is usually provenon a case-by-case basis for each different construction. The recently developedframework of parabolic molecules, which includes all known anisotropic frameconstructions based on parabolic scaling, provides a unified concept for a sparseapproximation results of such systems.

In this talk we will introduce the novel concept of α-molecules which allowsfor a unified framework encompassing most multiscale systems from the area ofapplied harmonic analysis with the parameter α serving as a measure for thedegree of anisotropy. The main result essentially states that the cross-Gramianof two systems with the same degree of anisotropy exhibits a strong off-diagonaldecay. One main consequence we will discuss is that all such systems thenshare similar approximation properties, and desirable approximation propertiesof one can be deduced for virtually any other system with the same degree ofanisotropy.

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A nonlinear derivative defined in the discrete domain and its appli-cationsOlivier LALIGANTLe2i Lab., University of [email protected]

A nonlinear derivative is directly defined in the discrete domain. This deriva-tive is motivated by the asymmetry of pattern in discrete signal as step or edgein 2D signal. Thanks to the special definition (in the discrete domain) of thisderivative, pattern can be detected in a univocal way. This derivative is theonly one able to perfectly detect and localize ideal edges in image. Beside thisfundamental benefit, the derivative has the nice property to reduce noise. Ap-plications to edge detection, noise reduction and noise estimation are describedand their performances are studied.

Shearlet-based analysis of singularities and applications to fluorescentimage analysis of neuronal culturesDemetrio LabateUniversity of [email protected]

Coauthors: Kanghui Guo, Burcin Ozcan, Manos Papadakis

During the last decade, a new generation of multiscale representations hasemerged - most notably the shearlet representation - offering a very powerfulframework for microlocal analysis. Using the shearlet transform, in particular, itis possible to derive a precise geometric characterization of the set of singularitiesof a large class of multidimensional functions and distributions. These propertiesprovide the theoretical underpinning for several innovative algorithms for imageprocessing and feature extraction. We will show an application of these ideasto the automated extraction of morphological features from fluorescent imagesof neuronal cultures.

The Two Weight Inequality for the Cauchy transform for R to C+

Michael T Lacey207 Mimosa [email protected]

Coauthors: E. Sawyer, C.-Y. Shen, Uriate-Tuero and B. Wick

We characterize those pairs of weights, on on the real line, and the otheron the complex plane, such that the Cauchy transform is bounded from L2 ofthe first weight, to L2 of the second. The characterization is in terms of ajoint Poisson A2 condition, and a suite of testing inequalities. This verifes aconjecture of Nazarov-Treil-Volberg.

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Applications of the inequality include a question of Donald Sarason on thecomposition of Toeplitz operators, and another of William Cohn, on the char-acterization of Carleson measures for model spaces.

On prolate shift frames and Shannon samplingJoseph LaekyNew Mexico State [email protected]

Coauthors: Jeffrey A. Hogan

Prolate spheroidal wave functions, for fixed time and bandwidth parame-ters, are eigenfunctions of time-limiting and band limiting, and they form anorthonormal basis for the Paley-Wiener space of the given bandwidth. We showthat, when suitably normalized, certain collections of their lattice shifts alsoform frames for the Paley-Wiener space. A collection of prolate shifts can alsoprovide a Riesz basis when normalized such that there is one prolate-shift perunit time-bandwidth. We use such shifts to help explain that, at least for certainsubspaces of the Paley-Wiener space, finite subseries of the Shannon samplingexpansion provide effective approximations of the full Shannon sampling expan-sion.

Spectral measures associated with Factorization of Lebesgue mea-suresChun-Kit LaiMcMaster [email protected]

Coauthors: Jean-Pierre Gabardo

Let Q be a fundamental domain of some full-rank lattice in Rd and let µand ν be two positive Borel measures on Rd such that the convolution µ ∗ν is a multiple of χQ. We consider the problem as to whether or not bothmeasures must be spectral (i.e. each of their respective associated L2 spaceadmits an orthogonal basis of exponentials) and we show that this is the casewhen Q = [0, 1]d. This theorem yields a large class of examples of spectralmeasures which are either absolutely continuous, singularly continuous or purelydiscrete spectral measures. In addition, we propose a generalized Fuglede’sConjecture for spectral measures on R1 and we show that it implies the classicalFuglede’s Conjecture on R1.

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Diagonalizing the finite Zak transform and the finite Balian-Low the-oremMark [email protected]

We give a diagonalization of the matrix representation of the finite Zaktransform and use it to produce a fractional finite Zak transform. We proceedto use the finite Zak transform to provide a conjecture for a quantitative finiteBalian-Low theorem. More specifically, we propose a minimzer of the Heisenbergsum which still yields an orthonormal basis under time-frequency shits in thefinite setting.

Oversampling of wavelet frames for real dilationsJakob LemvigTechnical University of [email protected]

Coauthors: Marcin Bownik

Oversampling of wavelet frames has been a subject of extensive study byseveral researchers dating back to the early 1990s. The first oversampling resultsare due to Chui and Shi (1994), who proved that oversampling by odd factorspreserves tightness of dyadic affine frames. This is now the central result of thesubject known as the Second Oversampling Theorem. In this talk, we generalizethe Second Oversampling Theorem for wavelet frames and dual wavelet framesin higher dimensions from the setting of integer dilations to real dilations.

Robust and Fast Subspace RecoveryGilad LermanUniversity of [email protected]

Coauthors: Work 1: Teng Zhang; Work 2: Michael McCoy, Joel Tropp, TengZhang; Work 3: Matthew Coudron; Work 4: John Goes, Teng Zhang, RamanArora; Work 5: Tyler Maunu

Consider a dataset of vector-valued observations that consists of a modestnumber of noisy inliers, which are explained well by a low-dimensional subspace,along with a large number of outliers, which have no linear structure. We firstdescribe a convex optimization problem that can reliably fit a low-dimensionalmodel to this type of data. When the inliers are contained in a low-dimensionalsubspace we provide a rigorous theory that describes when this optimizationcan recover the subspace exactly. We present an efficient algorithm for solvingthis optimization problem, whose computational cost is comparable to that

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of the non-truncated SVD. We also show that the sample complexity of theproposed subspace recovery is of the same order as PCA subspace recovery andwe consequently obtain some nontrivial robustness to noise. At last, we discussmodifications of this convex strategy to obtain some computational advantages.

Fast thresholding algorithms with feedbacks for sparse signal recoveryShidong LiDepartment of [email protected]

Coauthors: Tiebin Mi and Shidong Li

We provide another framework of fast iterative algorithms based on nullspace tuning, thresholding, and feedbacks for sparse signal recovery arising insparse representation and compressed sensing. Several algorithms with differ-ent feedbacks are derived. Convergence results are also provided. The corealgorithm is shown to converge in finite many steps under a (preconditioned)restricted isometry condition. Numerical studies about the effectiveness andthe speed of the algorithms are also presented. The algorithms are seen asparticularly effective for large scale problems.

Sampling and Inference for Spatiotemporal Single-Photon ImagingYue M. LuHarvard [email protected]

In this talk, we will present some of our recent work on spatiotemporal single-photon sensors (SPS). In particular, we will present the performance bounds ofthe SPS in acquiring light intensity fields; on time-sequential adaptive sensingschemes that allow one to push the imaging capabilities of SPS systems beyondthe nominal limit imposed by current hardware; and on new image formationalgorithms that can efficiently ”decode” the massive bitstreams generated bythe SPS.

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A Statistical Problem from Spectroscopy, whose Solution Hints ofCompressive SensingBradley LucierPurdue [email protected]

Coauthors: Gregery T. Buzzard (Purdue University), Dor Ben-Amotz, PurdueUniversity (Purdue University)

A number of Raman spectroscopy instruments incorporating micromirrorarrays have been built in Dor Ben-Amotz’s laboratory in the Chemistry De-partment at Purdue.

A question arises: How best to set up the mirrors to distinguish among agiven set of chemical species.

The answer comes from a non-standard problem in Optimal Design of Ex-periments, a subfield of statistics.

We’ll present an overview of the problem, the mathematical challenges, andsome experimental results.

Multiscale Geometric Methods for Statistical Learning and Data inHigh-DimensionsMauro MaggioniDuke [email protected]

Coauthors: M. Crosskey, S. Gerber, D. Lawlor, S. Minsker, N. Strawn

We discuss a family of ideas, algorithms, and results for analyzing variousnew and classical problems in the analysis of high-dimensional data sets. Thesemethods rely on the idea of performing suitable multiscale geometric decompo-sitions of the data, and exploiting such a decomposition to perform a variety oftasks in signal processing and statistical learning. In particular, we discuss theproblem of dictionary learning, where one is interested in constructing, given atraining set of signals, a set of vectors (dictionary) such that the signals admit asparse representation in terms of the dictionary vectors. We discuss a multiscalegeometric construction of such dictionaries, its computational cost and onlineversions, and finite sample guarantees on its quality. We then generalize part ofthis construction to other tasks, such as learning an estimator for the probabil-ity measure generating the data, again with fast algorithms with finite sampleguarantees, and for learning certain types of stochastic dynamical systems inhigh-dimensions.

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Multiscale High-Dimensional Neural Networks ApproximationsStphane MallatEcole Normale Suprieure, [email protected]

Coauthors: Xu Chen, Xiu-Yuan Cheng, Matthew Hirn

Learning problems are high-dimensional approximations which brutally facethe curse of dimensionality. Supervised and unsupervised learning require someform of dimensionality reduction, which amounts to computing invariants. Re-markable results have recently been obtained by highly non-linear deep neuralnetworks. We show that such approximation networks can be constructed withscale separation strategies, using iterated wavelet tranforms. For complex clas-sification problems, multiscale wavelet operators must be learned from data.Classification applications will be shown on images, sounds and unstructureddata.

Coorbit spaces with voice in a Frechet spaceFilippo De MariUniversity of Genova, [email protected]

Coauthors: Dahlke, Univ. Marburg, Germany De Vito, Univ. Genova, ItalyLabate, Univ. Houston, USA Steidl, Univ. Kaiserslautern, Germany Teschke,Univ. Neubrandenburg. Germany Vigogna, Univ. Genova, Italy.

This is joint work with Dahlke, De Vito, Labate, Steidl, Teschke and Vi-gogna. We set up a new general coorbit space theory for non integrable ker-nels. Given a unitary representation π of a group G on a Hilbert space H,we assume that it satisfies ‖f‖2 =

∫G|V f(x)|2 dx, whenever f ∈ H, where

V is the voice transform associated to an admissible vector u ∈ H, namelyV f(x) = 〈f, π(x)u〉H, and we suppose that the kernel K = V u belongs to T , arather general Frechet space of functions on the group that is required to satisfya number of axioms. This replaces the standard assumption K ∈ L1(G) andworks even when π is not irreducible.

This framework applies to the classical since kernel and to non-integrablewavelets such as the Shannon wavelet. Some of the most interesting exam-ples come from reproducing representations of triangular subgroups of Sp(2,R),including the so-called Schrodingerlets. In all these cases the natural Frechetspace is T =

⋂p∈(1,+∞) L

p(G). This theory is succesful in the sense that itprovides a workable sobstitute for the standard integrability condition on thekernel, it contains the classical coorbit space theory even for non irreduciblerepresentations, it applies to several interesting examples and it is compatiblewith the recent theory developed by Christensen and Olafsson.

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Some optimal design problems for finite framesPedro MasseyUniv. Nacional de La Plata and IAM-CONICET, [email protected]

Coauthors: Mariano Ruiz and Demetrio Stojanoff (Univ. Nacional de La Plataand IAM-CONICET, Argentina)

In this talk we will consider the following two optimal design problems infinite frame theory:

- Given a redundant frame F = fini=1 in Cd, we compute the spectraland geometrical structure of minimizers of the frame potential among the dualframes G = gini=1 for F such that

∑ni=1 ‖gi‖2 ≥ t and such that ‖TF#−TG‖ ≤

ε, where TG denotes the synthesis operator of G and F# denotes the canonicaldual of F .

- Given a frame F = fini=1 in Cd, we compute the spectral and geometricalstructure of the minimizers of the frame potential among all coherent pertur-bations V · F = V fini=1 where V is any invertible operator V such that‖V ∗V − I‖ ≤ δ and det(V ∗V ) ≥ s.

The motivation of these problems is the search of numerical stable encoding-decoding schemes based on (perturbations) of F . Our approach relies in Lidskii’stype inequalities (both additive and multiplicative) from matrix analysis theory.It turns out that the matrix models behind these two (seemingly unrelated)problems are intimately connected. The talk is based on joint work with MarianoRuiz and Demetrio Stojanoff.

Directional time-frequency analysis via continuous framesPeter MassopustTechnische Universitt [email protected]

Coauthors: Ole Christensen and Brigitte Forster

Using the concept of ridge functions, Grafakos and Sansing obtained di-rectionally sensitive time-frequency decompositions in L2(Rn) based on Gaborsystems in L2(R). We generalize their result by showing that similar results holdstarting with general frames for L2(R), both in the setting of discrete framesand continuous frames. This allows to apply the theory for several other classesof frames, e.g., wavelet frames and shift-invariant systems. We will considerapplications to the Meyer wavelet and complex B-splines. In the special case ofwavelet systems, we show how to discretize the representations using ε-nets.

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Stable sampling and Fourier MultipliersBasarab MateiLAGA Paris [email protected]

Coauthors: Y. Meyer, J. Ortega-Cerd

We study the relationship between stable sampling sequences for bandlim-ited functions in Lp(Rn) and the Fourier multipliers in Lp. In the case that thesequence is a lattice and the spectrum is a fundamental domain for the latticethe connection is complete. In the case of irregular sequences there is still apartial relationship.

Frames of translates on the Heisenberg groupAzita MayeliCity University of New [email protected]

Frames generated by translates of functions play an important role in sam-pling theory and wavelet theory. In this talk, we will give a complete charac-terization of frames and Riesz bases of translations on the Heisenberg groupanalogues to the characterization in the Euclidean case. More precisely, we willgive a necessary and sufficient condition for Plancherel transform of a functionwhose translations form a frame (resp. Riesz basis) for its closed linear span.

Memoryless scalar quantization (MSQ) for random framesKateryna MelnykovaUniversity of British [email protected]

Coauthors: Ozgur Yilmaz

Frames are known to provide stable and robust discrete signal representa-tions. In general, these representations are in terms of frame coefficients thatgenerally take on a continuous range of values. To be amenable to digital stor-age, transmission, and processing, it is crucial that the amplitudes of the framecoefficients are discretized or ”quantized”. The simplest quantization methodin this setting is memoryless scalar quantization (MSQ) where one essentiallyrounds off each frame coefficient separately. For error analysis of MSQ, the en-gineering literature often relies on the ”White Noise Hypothesis” (WNH) thatsays that, for a fixed frame and random signal, the coefficient quantization er-rors associated with each coefficient are i.i.d. random variables. Nonetheless,the WNH is shown to be not rigorous and is not completely valid, at least in cer-tain cases. In this talk, we will focus on Gaussian random frames and estimate

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the MSQ error rigorously. Specifically, we will show that the result essentiallycoincides with the error bound predicted by the WNH.

Simple n-dimensional wavelet setsKathy MerrillColorado [email protected]

New examples of wavelet sets that are finite unions of convex polytopes willbe presented for both scalar and matrix dilation in L2(Rn), for all n ≥ 2. Severalexamples with all faces parallel to coordinate planes will be included.

Nonlinear Dimensionality Reduction: The Inverse Map.Francois MeyerUniversity of Colorado at [email protected]

Coauthors: Nathan Monnig and Bengt Fornberg

Nonlinear dimensionality reduction embeddings computed from datasets donot provide a mechanism to compute the inverse map. In this talk, we addressthe problem of computing a stable inverse map to such a general bi-Lipschitzmap. The approach relies on radial basis functions (RBFs) to interpolate theinverse map everywhere on the low-dimensional image of the forward map. Wedemonstrate that the scale-free cubic RBF kernel performs better than the Gaus-sian kernel: it does not suffer from ill-conditioning, and does not require thechoice of a scale. The proposed construction is shown to be similar to the Nys-trom extension of the eigenvectors of the symmetric normalized graph Laplacianmatrix. Based on this observation, we provide a new interpretation of the Nys-trom extension with suggestions for improvement.

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Representation of functions on big data: graphs and treesH. N. MhaskarCalifornia Institute of Technology & Claremont Graduate [email protected]

Coauthors: C. K. Chui, F. Filbir

Many current problems dealing with big data can be cast efficiently as func-tion approximation on graphs. The information in the graph structure can oftenbe reorganized in the form of a tree; for example, using clustering techniques.The objective of this paper is to develop a new system of orthogonal functionson weighted trees. The system is local, easily implementable, and allows forscalable approximations without saturation. A novelty of our orthogonal sys-tem is that the Fourier projections are uniformly bounded in the supremumnorm. We describe in detail a construction of wavelet–like representations andestimate the degree of approximation of functions on the trees.

Accuracy, Sum Rules and Crystal WaveletsUrsula MolterFCEyN, Universidad de Buenos [email protected]

Coauthors: Maria del Carmen Moure, Alejandro Quintero

The scaling function of a crystal Multiresolution Analysis can be identifiedwith a vector-scaling function for a Multiresolution Analysis associated to alattice. Therefore, the well known techniques for this case can be applied.(Eg: determining the accuracy, approximation properties, etc.) However, thestructure that the underlying crystal group provides, enables us to simplifythese properties - and for example allows to find sum rules for multiwavelets,which before were only available for single functions. The group structure isalso helpful for the actual construction of crystal scaling functions. Further,when trying to build the crystal wavelet basis, much fewer calculations have tobe performed by taking advantage of the tight structure imposed by the crystalgroup.

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On structural decompositions of finite framesSivaram K. NarayanDepartment of Mathematics, Central Michigan University, Mount Pleasant, [email protected]

Coauthors: Alice Chan, Martin S. Copenhaver, Logan Stokols, Allison Theobold.

A frame in an n-dimensional Hilbert space Hn is a possibly redundant col-lection of vectors fii∈I that span the space. A tight frame is a generalizationof an orthonormal basis. A frame fii∈I is said to be scalable if there exist non-negative scalars cii∈I such that cifii∈I is a tight frame. In this report westudy the combinatorial structure of frames and their decomposition into tightor scalable subsets by using partially-ordered sets (posets). We define the factorposet of a frame fii∈I to be a collection of subsets of I ordered by inclusionso that nonempty J ⊆ I is in the factor poset if and only if fjj∈J is a tightframe for Hn. We prove conditions which factor posets satisfy and use these tostudy the inverse factor poset problem, which inquires when there exists a framewhose factor poset is some given poset P . We determine a necessary conditionfor solving the inverse factor poset problem in Hn which is also sufficient forH2. We then turn our attention to scalable frames and present partial resultsregarding when a frame can be scaled to have a given factor poset.

Combining Riesz bases in higher dimensionsShahaf NitzanKent State [email protected]

Coauthors: Gady Kozma

In a previous work, Joint with Gady Kozma, we proved that every finiteunion of intervals admits a Riesz basis of exponentials. In this talk we willdiscuss an extension of this result to higher dimensions: Every finite union ofrectangles, with edges parallel to the exes, admits a Riesz basis of exponentials.

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Approximating scalable framesKasso OkoudjouUniversity of [email protected]

Coauthors: Xuemei Chen, Gitta Kutyniok, Friedrich Philipp, and RongrongWang

A frame for a finite dimensional real Euclidean space is scalable if its vec-tors can be rescaled in such away that the resulting system becomes a tightframe. In the first part of this talk we shall present some quantitative measuresof scalability of a frame. The second part of the talk will focus on the bestapproximations of non scalable frames by scalable ones. The results we presentin this second part use in an essential way the previously introduced measuresof scalability.

Coorbits, function spaces and atomic decompositionGestur OlafssonLouisiana State [email protected]

Coauthors: J. Christensen, K.-H. Grochenig, A. Mayeli

Function spaces often come with natural symmetries and group action. Thisis one of the starting point in the Coorbit theory of Feichtinger and Grochenig.The theory was initiated in the late 1980’s and then developed further by severalauthors. We will give an overview over the ideas behind the coorbit theory,discuss new developments and examples.

Using p-adic MRA’s to analyze equivalence bimodules between non-commutative solenoidsJudith A. PackerUniversity of Colorado, [email protected]

Let p be a prime number, and consider a noncommutative solenoid C∗(Z[ 1p ]×

Z[ 1p ],Ψα) = Aα where Ψα is a multiplier on Z[ 1

p ]× Z[ 1p ]. The speaker together

with F. Latremoliere constructed a Morita equivalence bimodule between Aαand Aβ for a different multiplier Ψβ on Z[ 1

p ] × Z[ 1p ] by using a Heisenberg

equivalence bimodule construction due to M. Rieffel. This bimodule involvedcompleting continuous functionw with compact support on the locally compactabelian group M = [Qp × R]. This talk will use p-adic multiresolution analysesof Shelkovich and Skopina to construct the corresponding Hilbert module Ξbetween Aα and Aβ as a projective multiresolution structure.

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A Novel Look at Extension PrinciplesManos PapadakisUniversity of [email protected]

Coauthors: Nikolaos Atreas and Theodoros Stavropoulos

Extension Principles address a fundamental problem in wavelet constructionapplicable to digital data processing. When the integer translates of a refinablefunction do not form a frame for their closed linear span but form only a Besselfamily the construction of affine wavelet frames with desirable spatial localiza-tion cannot be carried out as in the classical multiresolution theory of Mallatand Meyer. In this case Extension Principles provide the complete answer tothis problem and prescribe an efficient design strategy for stable wavelet filterswith symmetry and antisymmetry properties and small support, because refin-able functions can be designed to have arbitrarily smooth Fourier transforms,compact support and nice symmetry properties in the spatial domain. We willpresent the most recent work on Extension Principles including results on exten-sion principles on distributional refinable functions. Our talk will highlight howany pair of homogeneous dual multiwavelet affine frames of L2(Rs) gives riseto a pair of inhomogeneous dual affine multiwavelet frames constructed from apair of refinable function vectors and vice versa. We will also show how thisequivalence between these two types of affine wavelet frames extend the MixedOblique Extension Principle.

Linear combinations of frame generators in systems of translatesVictoria PaternostroTechnische Universitt [email protected]

Coauthors: Carlos Cabrelli (University of Buenos Aires) Carolina Mosquera(University of Buenos Aires)

A finitely generated shift invariant space V is a closed subspace of L2(Rd)that is generated by the integer translates of a finite number of functions. A setof frame generators for V is a set of functions whose integer translates form aframe for V . In this talk we give necessary and sufficient conditions in order thata minimal set of frame generators can be obtained by taking linear combinationsof the given frame generators. Surprisingly the results are very different to therecently studied case when the property to be a frame is not required.

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Greedy algorithm for subspace clustering from corrupted and incom-plete dataAlexander PetukhovUniversity of [email protected]

Coauthors: I.Kozlov

We describe the Fast Greedy Sparse Subspace Clustering (FGSSC) algo-rithm providing an efficient method for clustering data belonging to a few low-dimensional linear or affine subspaces. The main difference of our algorithmfrom predecessors is its ability to work with noisy data having a high rate oferasures (missed entries at the known locations) and errors (corrupted entriesat unknown locations).

The algorithm has significant advantage over predecessor on synthetic mod-els as well as for the Extended Yale B dataset of facial images. In particular, theface recognition misclassification rate turned out to be 6–20 times lower than forthe SSC algorithm. FGSSC algorithm is able to perform clustering of corrupteddata efficiently even when the sum of subspace dimensions significantly exceedsthe dimension of the ambient space.

A Balian-Low theorem for subspaces of L2(R)Gotz PfanderJacobs [email protected]

Coauthors: Carlos Cabrelli, Ursula Molter

The Balian-Low theorem states that the window function ϕ of a Gabor Rieszbasis (ϕ, aZ×bZ) for L2(R cannot be well-localized in time and in frequency. Weextend this result to Gabor Riesz bases of subspaces of L2(R) that are invariantunder a time-frequency shift λ that lies outside the lattice a aZ× bZ.

Adaptive Stable Reconstruction of M-Sparse Vectors from FourierDataGerlind PlonkaUniversity of [email protected]

We propose a deterministic stable algorithm for sparse vector reconstructionfrom Fourier data. Particularly, if the vector N-dimensional vector x is M -sparse, we need at most minM log(N), N Fourier values in order to recover x.The algorithm works iteratively and does not incorporate any a priori knowledgeon the sparsity M of x. Each iteration step only involves the solution of a linearsystem of size at most M . If we are allowed to choose the Fourier samples for

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reconstruction adaptively at each iteration level then we can develop a strategyto ensure that the coefficient matrix in the linear system is well-conditioned.

The proposed method can be generalized to functions that can be sparselyrepresented in a finite basis.

Our idea is compared with the Prony method for vector reconstruction,where M -sparse vectors can theoretically be recovered by 2M Fourier data.However, the Prony method is particularly for larger M severely ill-conditioned.Further, we compare the approach with compressive sensing algorithms, wherethe Fourier data are chosen randomly, and where usually at least O(M logN)data are needed to obtain a correct reconstruction with high probability. Thetalk is based on joint work with Shai Dekel.

Adaptive total variation regularization in image processingSurya PrasathUniversity of [email protected]

We consider an adaptive version of the well-known total variation (TV)based regularization model for image restoration in image processing. AdaptiveTV based schemes try to avoid the staircasing artifacts [Nikolova, Local stronghomogeneity of a regularized estimator, SIAM J. Applied Math. 2000] associ-ated with TV regularization based energy minimization models. An algorithmbased on a modification of the split Bregman technique proposed by [Goldsteinand Osher, The split Bregman algorithm for L1 regularized problems, SIAMJ. Imaging Sci. 2009], can be used for solving the adaptive case. Convergenceanalysis of such an alternating scheme is proved using the Fenchel duality anda recent result on the weak convergence of Douglas-Rachford splitting method[Svaiter, On weak convergence of the Douglas-Rachford method, SIAM J. Con-trol Optim. 2011]. We demonstrate comparative experimental results usingthe modified split Bregman, dual minimization and additive operator splittingfor the gradient descent scheme for the TV diffusion equation to highlight theefficiency of adaptive TV based schemes for image denoising, restoration anddecomposition problems.

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Interpolation via compressive sensingHolger RauhutRWTH Aachen [email protected]

Coauthors: Rachel Ward, Christoph Schwab

Compressive sensing predicts that sparse vectors can be recovered from in-complete linear (randomized) measurements via efficient algorithms such as `1-minimization. This principle can be easily adapted to interpolation problemswhere the functions of interest are known to be well-approximated by a sparseexpansion in terms of an orthogonal function system such as the trigonometricsystem. In certain situations of interest, however, functions are not only sparsebut also smooth. It turns out that a good model of smooth approximately sparsefunctions is provided by a norm ball in a weighted `p-space on the expansioncoefficients with p < 1. In this context, one passes to weighted `1-minimizationas recovery method. We will present theoretical estimates on the achievableapproximation rates. Numerical experiments show the effectiveness of the newapproach. We also present an application to uncertainty quantification, i.e., thenumerical solution of parametric partial differential equations.

Big Data, Sigtetics IBC, and the Holomorphic Characterization ofQuantum Imaging Communication ChannelsDomingo RodriguezUniversity of Puerto Rico, [email protected]

This work formulates a type of quantum imaging communication channel,in the context of theoretical quantum detection and estimation operations per-formed in configuration or phase space, following some features of the workof Masanori Ohya on information dynamics. A quantum channel is describedin this work as a linear mapping between complex separable Hilbert spaces.A quantum communication channel, in turn, is described as a class of quan-tum channels, with an associated subclass of imaging channels. This work alsodiscusses an information-based complexity (IBC) approach, following the fun-damental work of Joseph F. Traub, for estimating a type of cost incurred whentrying to model this class of imaging channels under a discrete-time, discrete-frequency computational structure. It also tries to estimate the complexityassociated with the processing of big data generated through approximate so-lutions for potential quantum sensing applications. Finally, this work exploreshow the use of signal and tensor analytics (SIGTETICS) in a Toeplitz algebracomputational framework contributes to a discussion of an interpretation of cer-tain aspects of the Segal-Bargmann holomorphic representation, following theworks of Brian C. Hall and Peter Woit.

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The optimal transport transform and its application to pattern recog-nition in image databasesGustavo K. RohdeCarnegie Mellon [email protected]

Numerous applications in science and technology require pattern recogni-tion to be performed in signal and image databases. The predominant scenarioin these applications is that the dimension of available signals is usually muchlarger than the number of data samples available. In observing the variationsencountered in signal and image databases, it is important to note that thesecan be explained by changes in the signal intensities, as well as their locations.Inspired by the continuity equation, we describe a new signal processing frame-work that adopts a Lagrangian point of view (as opposed to the usual linearEulerian one) using concepts from optimal transport theory. Based on a lin-earized version of the optimal transport metric we have derived a new nonlinearsignal analysis and synthesis framework (i.e. a transform) that can be usefulin analyzing the structure of signal variations encountered in image databases.Preliminary theoretical and experimental evidence pointing to potential increasein classification accuracies in discrimination problems will be shown. Examplesusing image databases of faces and cells will be shown. Applications in cancerdetection (cytopathology) from images of cells will be presented.

Asymptotics for optimal spectrogramsJose Luis RomeroUniversity of [email protected]

Coauthors: Luis Daniel Abreu, Karlheinz Groechenig

We consider the problem of optimizing the concentration of the spectrogramof a function within a given target region and obtain asymptotics for the time-frequency profile of the corresponding solutions. The main result shows thatthese solutions are organized in such a way that they almost tile the targettime-frequency spectrum.

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Parseval quasi-dual framesMariano RuizUniv. Nac. de La Plata and Inst. Arg. de Matemtica [email protected]

Coauthors: Pedro Massey and Gustavo Corach

Let fii∈I and gii∈I be a pair of frames for a Hilbert space H, withsynthesis operators F and X respectively . In this talk, we discuss some resultsconcerning the computation of the infimum of the operator norm of FX∗ − I,where gii∈I is any Parseval frame. In some sense, this quantity measures the(normalized) worst-case error in the reconstruction of vectors when analyzedwith the Parseval frame and synthesized with fii∈I . In case that the infimumis attained on a Parseval frame gii∈I , we call it a quasi-dual of fii∈I . Thistalk is based on a joint work with Pedro Massey (UNLP-IAM, Argentina) andGustavo Corach (UBA-IAM, Argentina).

Random encoding of quantized finite frame expansionsRayan SaabMathematics Department, [email protected]

Coauthors: Mark Iwen

Frames generalize the notion of bases and provide a useful tool for modelingthe measurement (or sampling) process in several modern signal processing ap-plications. In the digital era, such a measurement process is typically followedby a quantization, or digitization step. One family of quantization methods,popular for its robustness to errors and ability to act progressively on the mea-surements is Sigma-Delta quantization. In this talk, we show that a simple post-processing step consisting of a discrete random embedding of the Sigma-Deltabit-stream yields near-optimal rate distortion performance with high probabil-ity, while allowing efficient reconstruction. Our result holds for a wide varietyof frames, including smooth frames and random frames.

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The framework of α-moleculesMartin SchferTechnische Universitt [email protected]

Coauthors: Philipp Grohs (Eidgenssische Technische Hochschule Zrich), SandraKeiper (Technische Universitt Berlin), Gitta Kutyniok (Technische UniversittBerlin)

Since classical wavelet systems are not optimally suited to represent multi-variate data if anisotropic structures are involved, such as edges or rays in im-ages for example, many new representation systems beyond wavelets have beendeveloped over the last decade. The considered model situation are functionswith singularities along lower dimensional embedded manifolds, with the aimto provide optimally sparse approximation of these objects. Some of the mostwell-known nowadays termed directional representation systems are ridgelets,curvelets, and shearlets, to name just a few.

The great variety of new systems motivated the search for a common frame-work with the ability to simultaneously establish general results, concerning forexample approximation properties. The concept of parabolic molecules, intro-duced recently by two of the collaborators, was a first step in this direction. Itcan unify shear-based and rotation-based constructions, such as classical shear-lets and curvelets, under one roof and allows to investigate their approximationbehavior simultaneously.

However, since the framework is confined to parabolic scaling, it cannotcomprise systems based on different scaling laws, like ridgelets, wavelets, ornewer hybrid constructions. This motivated the generalization of the conceptto α-molecules, where a more general anisotropic scaling law, specified by theparameter α, is utilized. This framework, which we will present in this talk,is general enough to comprise all the aforementioned constructions, and at thesame time still specific enough to capture their main features and properties.As a main result, we will show, that the framework of α-molecules can beused to identify large classes of representation systems with the same sparseapproximation behavior.

1-Bit Compressive Sensing: Models and AlgorithmsLixin ShenSyracuse [email protected]

One-bit compressive sensing models an extreme quantization in signal pro-cessing and recently came to the forefront of the field of compressive sensing.A number of efficient algorithms were developed recently for this problem. Inthis talk, we shall propose a new model and an algorithm for restoring sparsesignals from their 1-bit measurements.

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Multiresolution analyses through low-pass filter on local fields of pos-itive characteristicNiraj K. ShuklaIndian Institute of Technology Indore, Indore, [email protected]

Coauthors: Aparna Vyas

The concepts of a first-stage wavelet basis can be generalized to a countablesubset of a local field having positive characteristic by using a prime element ofsuch a field. In this paper, we provide a characterization of first-stage discretewavelet system on a countable subset of a local field of positive characteristic.Further, with the help of a first-stage wavelet basis, we get some results onrefinement equation and refinement coefficients which provides a constructionof multiresolution analysis on local field of positive characteristic.

Linear independence of the system of translates and uniqueness oftrigonometric seriesIvana SlamicUniversity of Rijeka, [email protected]

Given a square integrable function ψ, various properties of the correspond-ing system of integer translates can be expressed in terms of the periodizationfunction pψ. Various non-redundancy notions can be considered, and some ofthem have already been characterized. Namely, we know that `2-linear inde-pendence of the system is equivalent to periodization function being positivealmost everywhere, whilst the system is minimal if and only if 1/pψ belongs toL1.

We have considered the problem of `p-linear independence, where p 6= 2, inorder to find neccessary and sufficient conditions in terms of pψ. First, consider-ing p < 2 case, we have established the connection with `p-sets of uniqueness –the sets of positive Lebesgue measure that do not support functions with Fouriercoefficients in `p and were first introduced by Y.Katznelson.

Here, we present the results on the case where p > 2. The results on `2-linearindependence and minimality were leading us naturally to several conjectures,which appeared to be false. We present a method of construction of linearlydependent systems which can be considered as counterexamples to such conjec-tures. Various results concerning properties of the partial sums of trigonometricnull series play part in this construction. On the other hand, a new sufficientcondition can be given in terms of sets of uniqueness for trigonometric series,despite such sets can be found amongst sets of Lebesgue measure zero. More-over, adding an assumption that concerns Rajchman’s multiplication theorem,we can give a characterization of `p-linear independence.

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Admissible frames and Fourier frame approximationGuohui SongClarkson [email protected]

Coauthors: Anne Gelb

Frames can be seen as a generalization of basis with over-completeness. Lo-calized frames have obtained popularity and demonstrated robustness and effi-ciency in extensive literature of sampling. There are still some applications inwhich the data are sampled with weakly-localized frames. We will introduceadmissible frames that could be used to obtain stable and accurate reconstruc-tion with sampling data from weakly-localized frames. In particular, we applyit to Fourier frames and present fast and stable algorithms for Fourier frameapproximation.

Nonnegativity as an obstruction to spanning structureAnneliese H. SpaethHuntingdon [email protected]

Coauthors: Alexander M. Powell

Many obstructions to spanning structure are widely studied, including time-frequency localization and translation invariance. We examine the obstructionof pointwise nonnegativity of functions in Lp(R) to spanning structure in thosespaces. Some structures for which we have answered the existence questioninclude monotone bases, unconditional bases, unconditional quasibases, Rieszbases, frames, Markushevich bases, and conditional quasibases.

Linear independence of time-frequency shifts of functions with fastdecayDarrin SpeegleSaint Louis [email protected]

Coauthors: Marcin Bownik

The HRT conjecture states that time-frequency shifts of L2 functions arelinearly independent.

In this talk, we establish the linear independence of time-frequency translatesfor functions f having one sided decay limx→∞ |f(x)|ecx log x = 0 for all c > 0.Generalizations to higher dimensions are also considered.

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Total frame potential and its applications in data clusteringTobias SpringerTU [email protected]

Coauthors: Joachim Stockler, Katja Ickstadt

For statistical analysis of microarray gene expression data, the clustering ofshort time series is an important objective in order to identify subsets of genessharing a temporal expression pattern. An established method, the Short TimeSeries Expression Miner (STEM) by Ernst et al. (2005), assigns time seriesdata to the closest of suitably selected prototypes followed by the selection ofsignificant clusters and eventual grouping.

For the clustering of normalized d-dimensional data Y = yjj=1,...,N wepropose to minimize the penalized frame potential

Fα(Θ) =1

αTFP(Θ) −

m∑`=1

maxj=1,...,N

〈yj , θ`〉 (3)

for α > 0. The functional contains the “total frame potential” TFP of FiniteUnit Norm Tight Frames (FUNTFs), see Benedetto and Fickus (2003), andincludes a data-driven component for the selection of prototypes. We show thatthe solution of the corresponding constrained optimization problem is naturallyconnected to the spherical Dirichlet cells

Dj = v ∈ Rd : ‖v‖2 = 1, 〈yj , v〉 = maxk=1,...,N

〈yk, v〉

of the given normalized data. Furthermore, the minimizers of Fα are, giventhat α > 0, in the interior of the Dirichlet cells and the objective function Fαis differentiable in the minimum with the extremal condition

4TT ∗T + 2TΛ = αYs

where T , Ys ∈ Rd have normalized columns and Λ = diag(λ1, . . . , λm) containsthe Lagrange multipliers.

The general problem is closely related to the search for point configurationson the unit sphere like in Tammes’ (1930) or Thomson’s Problem (1904). More-over, the minimization of (??) (subject to the constraint that the solution isnormalized) contains connections to problems in matrix completion (see e.g.Candes and Tao (2009) or Mazumder, Hastie and Tibshirani (2010)).

The idea of using the frame potential in combination with a data-dependentterm for optimization was originally proposed by Benedetto, Czaja and Ehler(2010) for finding sparse representations.

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A Bayesian approach to the multivariate change-point problem in thewavelet domainRobert StewardSt. Louis [email protected]

We look at the statistical change-point problem in the multivariate setting.In particular, suppose we observe some stochastic process that undergoes a shiftto the process mean at an unknown time. We propose a multivariate method forpredicting this change-point location by conducting a Bayesian analysis on theempirical wavelet detail coefficients of the original time series. We show that ifthe mean function of our time series is expressed as a multivariate step function,then our Bayesian-wavelet method performs comparably with classical methodssuch as maximum likelihood estimation (MLE). The advantage to our method isseen in its ability to adapt to more general situations such as piecewise smoothmean functions.

Gabor Frames, Sampling Matrices, and Total PositivityJoachim StocklerTechnische Universitat Dortmund, [email protected]

Many properties of Gabor Frames G(g,Λ) on a regular time-frequency latticeΛ = αZ× βZ can be expressed in terms of the matrices

P (x) = (g(x+ αj − k/β))j,k∈Z, x ∈ R,

thanks to the results by Ron and Shen in the 1990’s and their fiberizationtechnique for shift-invariant systems. In particular, frame bounds for G(g,Λ) aredirectly obtained from uniform operator bounds of P (x) and operator bounds ofthe Moore-Penrose pseudo-inverses P †(x). We call matrices of the more generalform P = (g(xj − yk))j,k∈Z sampling matrices.

In my talk, I explain the relation of density results for Gabor frames andsampling matrices. The necessary condition αβ ≤ 1 for Gabor frames, in gen-eral, is closely related to density results for sampling in shift-invariant spaces.For special window functions g, namely if g is a totally positive function of finitetype, the sufficient condition αβ < 1 was found by K. Grochenig and the authorin [1], and was related to a sampling density for g.

In recent work with T. Kloos, we use the close connection of totally positivefunctions and exponential B-splines in order to study properties of the Gaborframe G(g,Λ). We show that Zak transforms of g have only one zero in theirfundamental domain of quasi-periodicity and, for a special example, we givegood estimates for the lower frame-bound near the critical density.

1. K. Grochenig, J. Stockler, Gabor frames and totally positive functions,Duke Math. J. 162(2013), 1003-1031.

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Invertible frame multipliers and their inversesDiana StoevaAcoustics Research Institute, Austrian Academy of [email protected]

Coauthors: Peter Balazs (Acoustics Research Institute, Austrian Academy ofSciences)

Frame multipliers are operators which combine analysis with one frame, mul-tiplication with a bounded scalar sequence (called the symbol), and synthesiswith (possibly) another frame. They are not only interesting from a theoreticalpoint of view, but also important for applications. In signal processing, multi-pliers are a particular way to implement time-variant filters. They are used forexample in denoising and psychoacoustical modeling.

In this talk we concentrate on invertible frame multipliers. When the twoframes are Riesz bases and the symbol is semi-normalized, one can invert thecorresponding multiplier using a multiplier with the reciprocal symbol and thecanonical duals of the given Riesz bases. Here we extend the class of Rieszbases to a larger class of frames, allowing the same way of inversion (using thereciprocal symbol and the canonical duals). For the general case of overcompleteframes, where such way of inverting might not work, we give a formula for theinverse multiplier using any dual of one of the frames and a unique dual of theother one. Further, we consider the case of Gabor multipliers. We give necessaryand sufficient conditions for an invertible operator on L2 (and its inverse) to berepresented as a Gabor frame multiplier with a constant symbol.

Discrete lines and geometric compressed sensingIrena StojanoskaTechnische Universitt [email protected]

Coauthors: Gitta Kutyniok (Technische Universitt Berlin)

Compressed sensing is a novel methodology in data processing which takesadvantage of the fact that most signals admit a sparse representation. In sucha case it then allows to recover the signal from considerably less measurementsthan those required by traditional methods.

We are interested in exploiting additional information about the signal,namely having sparse geometric structure. One goal in this setting is to im-prove the existing compressed sensing results over those where no structure isassumed. Another goal is to broaden the range of applications of the method-ology of compressed sensing.

In this work we present signals consisting of unions of discrete lines as thesimplest case of geometric sparsity. We discuss their properties and the applica-tion of compressed sensing to such signal models. Our results include construc-tion of a unit norm tight frame from the set of discrete lines; a weaker version ofthe restricted isometry property of our measurement matrix for sparse vectors

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with random support, as well as application of compressed sensing in separationof discrete points and lines.

The abc-problem for Gabor systemsQiyu SunUniversity of Central [email protected]

Coauthors: Xin-rong Dai

A long standing problem in Gabor theory is to identify ideal window func-tions on intervals of length c and time-frequency shift lattices aZ×bZ such thatthe corresponding Gabor system is a frame for the space of all square-integrablefunctions on the real line. In this talk, I will present our answer to the aboveabc-problem for Gabor systems.

Triangular subgroups of Sp(n,R) and reproducing formulaeAnita TabaccoPolitecnico di [email protected]

Coauthors: Elena Cordero

We consider the (extended) metaplectic representation of the semidirectproduct G = Hd o Sp(d,R) between the Heisenberg group and the symplec-tic group. Subgroups H = Σ oD, with Σ being a d× d symmetric matrix andD a closed subgroup of GL(d,R), are our main concern. We shall give a generalsetting for the reproducibility of such groups. As a byproduct, the extendedmetaplectic representation restricted to some classes of such subgroups is eitherthe Schrodinger representation of R2d or the wavelet representation of Rd oD,with D closed subgroup of GL(d,R). Finally, we shall provide new examples ofreproducing groups of the type H = Σ oD, in dimension d = 2.

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Nonlinear sparse reconstructions in Hilbert spacesTang, Wai-ShingNational University of [email protected]

We consider a new framework of nonlinear sparse reconstruction in a Hilbertspace setting, which may be thought of as nonlinear extension of finite-dimensionalsparse recovery problems to infinite-dimensional spaces. We establish expo-nential convergence of the iterative hard thresholding algorithm to reconstructsparse signals in a union of closed linear subspaces of a Hilbert space from theirnonlinear observations. We also propose an optimization framework in a Ba-nach space setting and use it to stably reconstruct signals in a union of closedlinear subspaces of a Hilbert space.

This research is joint work with Qiyu Sun.

Greedy algorithms in compressed sensingVladimir TemlyakovUniversity of South [email protected]

We study sparse representations and sparse approximations with respectto incoherent dictionaries. We address the problem of designing and analyz-ing greedy methods of approximation. A key question in this regard is: How tomeasure efficiency of a specific algorithm? Answering this question we prove theLebesgue-type inequalities for algorithms under consideration. A very impor-tant new ingredient of the talk is that we perform our analysis in a Banach spaceinstead of a Hilbert space. It is known that in many numerical problems users aresatisfied with a Hilbert space setting and do not consider a more general settingin a Banach space. There are known arguments that justify interest in Banachspaces. In this talk we give one more argument in favor of consideration ofgreedy approximation in Banach spaces. We introduce a concept of M -coherentdictionary in a Banach space which is a generalization of the corresponding con-cept in a Hilbert space. We analyze the Quasi-Orthogonal Greedy Algorithm(QOGA), which is a generalization of the Orthogonal Greedy Algorithm (Or-thogonal Matching Pursuit) for Banach spaces. It is known that the QOGArecovers exactly S-sparse signals after S iterations provided S < (1 + 1/M)/2.This result is well known for the Orthogonal Greedy Algorithm in Hilbert spaces.The following question is of great importance: Are there dictionaries in Rn suchthat their coherence in `np is less than their coherence in `n2 for some p ∈ (1,∞)?We show that the answer to the above question is yes. Thus, for such dictionar-ies, replacing the Hilbert space `n2 by a Banach space `np we improve an upperbound for sparsity that guarantees an exact recovery of a signal.

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Sparse stochastic processes and operator-like wavelet expansionsMichael UnserEPFL, Lausanne, [email protected]

We introduce an extended family of continuous-domain sparse processes thatare specified by a generic (non-Gaussian) innovation model or, equivalently, assolutions of linear stochastic differential equations driven by white Lvy noise.We present the functional tools for their characterization. We show that theirprobability distributions are infinitely divisible, which induces two distinct typesof behaviorGaussian vs. sparseat the exclusion of any other. This is the key toproving that the non-Gaussian members of the family admit a sparse represen-tation in a matched wavelet basis.

We use the characteristic form of these processes to deduce their transform-domain statistics and to precisely assess residual dependencies. These ideas areillustrated with examples of sparse processes for which operator-like waveletsoutperform the classical KLT (or DCT) and result in an independent componentanalysis. Finally, for the case of self-similar processes, we show that the wavelet-domain probability laws are ruled by a diffusion-like equation that describestheir evolution across scale.

Democracy of shearlet bases with applications to approximation andinterpolationDaniel VeraUniversity of [email protected]

Shearlets are based on wavelets with composite dilation and inherit impor-tant features from wavelets. Shearlets provide (near) optimal approximationfor the class of so-called cartoon-like images. Moreover, there are distributionspaces associated to them and there exist embeddings between these and clas-sical (dyadic isotropic) inhomogeneous spaces. We prove that the shearlets aredemocratic bases for the shear anisotropic inhomogeneous Besov and Triebel-Lizorkin spaces (i.e. they verify the p-Temlyakov property also known as p-spaceproperty) for certain parameters. Then, we prove embeddings (or characteri-zations) between approximation spaces and discrete weighted Lorentz spaces(in the framework of shearlet systems) and prove (that these embeddings areequivalent to) Jackson and Bernstein type inequalities. This allows us to find(real) interpolation spaces.

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Non integrable representations and coorbit space theoryStefano VigognaUniversit degli Studi di Genova, Dipartimento di [email protected]

Coauthors: Stephan Dahlke (Marburg), Filippo De Mari (Genova), Ernesto DeVito (Genova), Demetrio Labate (Houston), Gabriele Steidl (Kaiserslautern),Gerd Teschke (Neubrandenburg).

For a unitary representation π of a locally compact group G (with its Haarmeasure) on a Hilbert space H, the voice transform Vu : H → L∞(G) withrespect to a fixed vector u ∈ H is defined by Vuv(x) := 〈v, π(x)u〉. The repre-sentation π is called reproducing if Vu is an isometry of H into L2(G). In signalanalysis, reproducing representations perform very efficient reconstruction pro-cedures, as it is well established in many applications, notably for wavelets andshearlets. Coorbit space theory allows to extend the reproducing propertiesto a whole family of Banach spaces of functions and distributions, providing acharacterization of smoothness spaces in terms of the decay of the voice trans-form. Classically, this is possible under two basic assumptions: the reproducingrepresentation has to be irreducible, and the reproducing kernel Vuu has tobe integrable. I will show some examples of non irreducible representationswith non integrable kernels which motivate the interest for a generalization ofthe classical coorbit theory. As a particularly interesting case, I will exhibita new reproducing representation, whose admissible vectors are the so-calledSchrodingerlets. I will sketch the basic ideas that allow to overcome the clas-sical obstructions and suggest how a more general theory can naturally arise.The general theory will be described in detail in De Mari’s talk.

Phase Retrieval Using Bandlimited Window FunctionsAdityavikram ViswanathanMichigan State [email protected]

Phase Retrieval refers to the problem of reconstructing a signal from itsintensity measurements. It arises in many applications such as x-ray crystal-lography and diffraction imaging, where the detectors only capture intensityinformation about the underlying physical process. As is well known, the phaseencapsulates vital information about the underlying signal, which makes signalrecovery from such measurements extremely challenging.

In this work, we introduce a novel and efficient computational framework forreconstructing signals from their magnitude measurements. Through the use ofwindow functions (or masks), we estimate phase differences between pairs ofmagnitude measurements. The band-limited design of these window functionsallows us to restrict this computation to a small subset of all possible phase dif-ference pairs. We then solve an angular synchronization problem to recover the

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unknown phases. Theoretical and numerical results demonstrating the accuracyand efficiency of this reconstruction framework will be presented.

Compressive Sensing Based MIMO RadarHaichao WangUniversity of California, [email protected]

Coauthors: Thomas Strohmer

We derive a theoretical framework for the recoverability of targets in theazimuth-range-Doppler domain using tools developed in the area of compressivesensing. We will also consider the case that the targets are not assumed to beon the grids.

A one stage reconstruction method for Sigma-Delta quantization incompressed sensingRongrong WangUniversity of British [email protected]

Coauthors: Ozgur Yilmaz

We introduce an alternative reconstruction method for signals whose com-pressed samples are quantized via a Sigma-Delta quantizer. The method is basedon solving a convex optimization problem, and unlike the previous approaches,it is stable and robust and admits an unconditional convergence to the truesolution no matter whether the support is recovered or not. As a consequence,with an appropriate choice of quantization order, the method can achieve aroot exponential error decay with respect to the oversampling rate. Finally,our theory applies to ”fine” Sigma-Delta quantizers and ”coarse” Sigma-Deltaquantizers, e.g., 1-bit per measurement.

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Sub-linear Algorithm for Recovering Sparse Fourier SeriesYang WangMichigan State [email protected]

Coauthors: Andrew Christlieb, David Lawlor

We present new deterministic algorithms for the sparse Fourier transformproblem, in which we seek to identify k ≤ N significant Fourier coefficients froma signal of bandwidth N . Previous deterministic algorithms exhibit quadraticruntime scaling, while our algorithms scales linearly with k in the average casein the noiseless setting. We also present a multi-scale algorithm for noisy signalswhich proves to be extremely robust and efficient. This multi-scale algorithm isbased on the beta-expansion scheme for robust A/D conversion. We also presentthe first efficient algorithm for ultra-high dimensions signals.

Density Conditions for Sampling in de Branges SpacesEric WeberIowa State [email protected]

Coauthors: Sa’ud al-Sa’di, Hashemite University, Jordan

We consider the problems of sampling and interpolation in de Branges spacesHilbertspaces of entire functions which are square integrable on the real line with re-spect to some weight function and satisfy some growth conditions. The class ofde Branges spaces considered are those whose weight function has a phase func-tion which is bounded below. For this class, we prove that the HomogeneousApproximation Property holds for the reproducing kernel. As a consequence,necessary conditions for sampling and interpolating sequences are shown, whichgeneralize some well-known sampling and interpolation results in the Paley-Wiener space.

Beyond harmonic analysis with Fourier-like transformsCameron WilliamsUniversity of [email protected]

Coauthors: Bernhard G. Bodmann (University of Houston) Donald J. Kouri(University of Houston)

This talk presents a family of transforms which share many properties withthe Fourier transform. We prove that these transforms are isometries on L2(R)and enjoy the same scaling property. The transforms can be chosen to leavefunctions of Gaussian or super-Gaussian type invariant. We also establish short-time analogs of these transforms.

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Reconstruction of structured functions from sparse Fourier dataMarius WischerhoffInstitute for Numerical and Applied Mathematics, University of Gttingen, [email protected]

Coauthors: Gerlind Plonka

In this talk we will use the Prony method for parameter estimation of struc-tured functions to reconstruct some structured, real-valued functions by a smallnumber of Fourier samples.

First, we will consider the case of non-uniform translates of radial functions.In particular, we will show that linear combinations of N shifted versions ofa radial function, where the shifts and the coefficients are real-valued, can beuniquely recovered from 3N + 1 Fourier samples which are taken from threelines through the origin in the Fourier domain. Further, we will show that thisapproach can be generalized to the case of d-variate functions with d > 2.

In the second part of the talk, we will turn to the reconstruction of polygonalshapes, i.e., we will consider characteristic functions f = 1D of a domain D inthe real plane, where D is a polygon with N vertices. We will show that thepolygon D can be uniquely recovered from 3N samples of the Fourier transformof f , where these samples are taken from three lines through the origin in theFourier space.

Kernel-based Approximation Methods for Stchastic Partial Differen-tial EquationsQi YeDepartment of Mathematics, Syracuse [email protected]

In this talk, we will show how to solve the high-dimensional stochastic partialdifferential equations by the kernel-based approximaiton methods. We extendthe classical kernel-based approaches for deterministic data to stochastic data.The kernel-based methods are meshfree methods, and the kernel-based estima-tors are also feasible for high-dimensional domains or complicated boundaryconditions. The kernel-based approximate solutions of the stochastic equationsare constructed by the suitable positive definite kernels, e.g., compact supportkernels (Wendland functions) and Sobolev-spline kernels (Matern functions).

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Quantization of compressed sensing measurements: stability, robust-ness, and moreOzgur YilmazThe University of British [email protected]

In this talk, we will discuss how to efficiently quantize compressive sam-ples of sparse or compressible signals. Our focus will be Sigma-Delta quantiza-tion. Sigma-Delta methods are typically used to quantize redundant expansions,e.g., oversampled bandlimited functions or frame expansions. We have recentlyshown that these quantizers also provide superior approximations when used inthe compressed sensing setting by establishing a link between compressed sens-ing quantization and frame quantization. Our original result was only valid inthe case of exactly sparse signals with no noise and with Gaussian measurementmatrices. I will summarize recent developments which remove these restrictiverequirements and further enable us to obtain approximation rates that are bet-ter than inverse polynomial in the number of measurements. The recent resultsare joint work with Rongrong Wang and Rayan Saab.

Two-distance tight framesWei-Hsuan YuUniversity of Maryland, College [email protected]

Coauthors: Alexander Barg, Kasso Okoudjou

A set of S of unit vectors in n-dimensional Euclidean space is called sphericaltwo-distance sets, if there are two numbers a and b so that the inner productsof distinct vectors of S either a or b. When a+ b 6= 0, we derive new structuralproperties of the Gram matrix of a two-distance set that also forms a tight framefor Rn. In addition, we establish a one-to-one correspondence between two-distance tight frames with certain strongly regular graph. This allows us to usethe adjacent matrix of these strongly regular graphs to construct two-distancetight frames. Several new examples are obtained along this characterization.

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Edge detection of brain tumor using curvelet transformNoore ZahraSharda Universitynoor zahra [email protected]

Coauthors: Shahin Naz, Manisha Rajoriya

Medical images has lots of curves, phase information and textured infor-mation. Edges are the most prominent feature for images. Edges defined theboundaries between different textures it shows discontinuities in image intensityfrom one pixel to other. In medical imaging detecting and enhancing the bound-aries between cavities is an important task and curvelet transform is proved asas a best tool for edge detection in the recent few years. The curvelet trans-form is a higher dimensional generalization of the wavelet transform designedto represent images at different scales and different angles. Curvelet transformcapture efficiently edges in an image; it is a good candidate for multiscale edgeenhancement. In this paper curvelet transform is used to find the edges in tu-mor. The idea is to modify the curvelet coefficients of the input image in orderto enhance its edges.

On the relationship between the Mittag-Leffler transform and thefractional Fourier transformAhmed ZayedDePaul [email protected]

There are different extensions of the Fourier transform to fractional orders.The most common extension in applications is based on the observation that theHermite functions are eigenfunctions of the Fourier transform. This extensionof the Fourier transform is related to a class of integral transforms known aslinear canonical transforms which play an important role in optics and signalprocessing.

On the other hand, there is another extension of the Fourier transform tofractional orders based on the Mittag-Leffler transform. This extension is relatedto fractional derivatives and Taylor’s series of fractional order.

In this talk we examine the connection between these extensions of theFourier transform and the sampling expansions associated with them.

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Tensor product complex tight framelet: construction and applicationZhenpeng ZhaoUniversity of [email protected]

Coauthors: Bin Han, and Qun Mo

In this talk, we shall address the construction of tensor product complex tightframelets with increasing directionality with respect to the real-valued wavelets.The applications of tensor product complex tight framelets on increasing thedirectionality of initial stage filters in dual tree complex wavelet transform andimage denoisnig with mixtures of Gaussian scale mixture model will also bediscussed. We shall show that tensor product complex tight framelets havesuperior performance in image denoising, compared with undecimated wavelettransform, and etc.

Mathematical analysis for information theoretic learningDing-Xuan ZhouCity University of Hong [email protected]

Information theoretic learning is a learning framework that uses descriptorsfrom information theory (entropy and divergence) estimated directly from datato substitute the conventional statistical descriptors of variance and covariance.Minimum error entropy (MEE) is a principle for designing supervised learningalgorithms that falls into the information theoretic learning framework. MEE al-gorithms have been applied successfully in various fields for more than a decade,and can deal with problems involving non-Gaussian noise for which the classi-cal least squares method is not ideal. In this talk we consider empirical MEElearning algorithms in a regression setting. Statistical consistency of an MEEalgorithm in an empirical risk minimization framework is presented in details,including error entropy consistency and regression consistency for homoskedas-tic models and heteroskedastic models. Fourier analysis plays a crucial rule inour analysis. Regularization schemes in reproducing kernel Hilbert spaces arealso discussed.

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Smooth affine shear tight framesXiaosheng ZhuangCity Univ. of Hong [email protected]

Coauthors: Bin Han

A directional wavelet tight frame is generated by isotropic dilations andtranslations of directional wavelet generators, while an affine shear tight frameis generated by anisotropic dilations, shears, and translations of shearlet gen-erators. We shall show that these two tight frames are actually connected inthe sense that the affine shear tight frame can be obtained from a directionalwavelet tight frame through sub-samplings. Consequently, the affine shear tightframe indeed has an underlying filter bank systems from the MRA structure ofthe directional wavelet tight frame.

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5th international conference on computational harmonic ...for the algebraic problem, we de ne a...

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